<PAGE>

                                  EXHIBIT 4.5
<PAGE>

NEITHER THIS WARRANT NOR THE SECURITIES ISSUABLE UPON EXERCISE HEREOF NOR ANY
INTEREST OR PARTICIPATION HEREIN OR THEREIN MAY BE SOLD, ASSIGNED, PLEDGED,
HYPOTHECATED, ENCUMBERED OR IN ANY OTHER MANNER TRANSFERRED OR DISPOSED OF
EXCEPT IN COMPLIANCE WITH THE SECURITIES ACT OF 1933, AS AMENDED, AND THE TERMS
AND CONDITIONS HEREOF.  THE HOLDER OF THIS WARRANT AND THE SECURITIES ISSUABLE
UPON EXERCISE HEREOF ARE SUBJECT TO THE RESTRICTIONS HEREIN SET FORTH.

VOID AFTER 5:00 P.M. NEW YORK CITY TIME, JANUARY 24, 2011


                   ****************************************


                                   No. W059

                                    WARRANT

                                      to

                             PURCHASE COMMON STOCK

                                      of

                              F.Y.I. INCORPORATED



                   ****************************************


          This certifies that, for good and valuable consideration, F.Y.I.
Incorporated, a Delaware corporation (the "Company"), grants to JOE A. ROSE or
permitted registered assigns (the "Warrantholder" or "Warrantholders"), the
right to subscribe for and purchase from the Company, at $30.375 per share (the
"Exercise Price"), fifty-eight thousand four hundred (58,400) shares of the
Company's Common Stock, par value $0.01 per share (the "Common Stock"), subject
to the provisions and upon the terms and conditions herein set forth.  The
Exercise Price and the number of Warrant Shares are subject to adjustment from
time to time as provided in Section 5.
<PAGE>

     1.     Duration and Exercise of Warrant; Limitation Exercise Payment of
            ----------------------------------------------------------------
            Taxes.
            -----

            1.1. Duration and Exercise of Warrant.
                 --------------------------------

     (a)  This Warrant may be exercised as to 100% of the underlying shares at
any time following the date of receipt by the Compensation Committee of the
Board of Directors of the Company's audited financial statements showing the
Company's actual earnings per share for the year ended December 31, 2001
("fiscal 2001"), provided that such earnings are not less than $2.28 per share
(subject to Section 1.1(d)).  Subject to Section 1.1(d), in the event the
Company's actual fiscal 2001 earnings are less than $2.28 per share, but at
least $2.23 per share, then this Warrant shall be exercisable as to the
percentage indicated across from the designated earnings per share target set
forth below:


Fiscal 2001 EPS Threshold                % of Warrant Exercisable
-------------------------                ------------------------


                  $2.23                            50%
                  $2.24                            60%
                  $2.25                            70%
                  $2.26                            80%
                  $2.27                            90%


The Company shall use its best efforts to deliver to the Compensation Committee
the audited financial statements showing the Company's actual fiscal 2001
earnings per share by March 5, 2002. However, in any case (and subject to
Section 1.2), this Warrant shall vest as to 100% of the underlying shares on
March 5, 2010.  The date this Warrant is first exercisable as to a portion of
the Warrant is hereinafter referred to as the "Exercise Date" for such portion
of the Warrant.  The Company shall give prompt notice to the Warrantholder of
the Exercise Date in accordance with Section 7.6.  This Warrant expires at 5:00
P.M., New York City time on January 24, 2011 (the "Expiration Date").  In
addition, in the event of a Change in Control of the Company, the right to
exercise 100% of the underlying shares shall immediately vest.  A "Change in
Control" shall be deemed to have occurred if:

          (i)  any person, other than the Company or an employee benefit plan of
     the Company, acquires directly or indirectly the Beneficial Ownership (as
     defined in Section 13(d) of the Securities and Exchange Act of 1934, as
     amended (the "Exchange Act")) of any voting security of the Company and
     immediately after such acquisition such Person is, directly or indirectly,
     the Beneficial Owner of voting securities representing 30% or more of the
     total voting power of all of the then-outstanding voting securities of the
     Company;

          (ii) the individuals (A) who, as of the closing date of the Initial
     Public Offering, constitute the Board (the "Original Directors") or (B) who
     thereafter are elected to the Board and whose election, or nomination for
     election, to the Board was approved by a vote of at least two-thirds (2/3)
     of the Original Directors then still in office (such directors becoming
     "Additional Original Directors" immediately following their election) or
     (C) who are elected to the Board and whose election, or nomination for
     election, to the

                                       2
<PAGE>

     Board was approved by a vote of at least two-thirds (2/3) of the Original
     Directors and Additional Original Directors then still in office (such
     directors also becoming "Additional Original Directors" immediately
     following their election), cease for any reason to constitute a majority of
     the members of the Board;

          (iii) the stockholders of the Company shall approve a merger,
     consolidation, recapitalization, or reorganization of the Company, a
     reverse stock split of the outstanding voting securities of the Company, or
     consummation of any such transaction if stockholder approval is not sought
     or obtained, other than any such transaction which would result in at least
     75% of the total voting power represented by the voting securities of the
     surviving entity outstanding immediately after such transaction being
     Beneficially Owned by the holders of at least 75% of the outstanding voting
     securities of the Company immediately prior to the transaction, with the
     voting power of each such continuing holder relative to other such
     continuing holders not substantially altered in the transaction; or

          (iv)  the stockholders of the Company shall approve a plan of complete
     liquidation of the Company or an agreement for the sale or disposition by
     the Company of all or a substantial portion of the Company's assets (i.e.,
     50% or more of the total assets of the Company).

     (b)  The rights represented by this Warrant may be exercised by the
Warrantholder of record, in whole, or from time to time in part, by (a)
surrender of this Warrant, accompanied by either the Exercise Form annexed
hereto, or if the Warrantholder decides to exercise the Warrant pursuant to the
broker-assisted cashless exercise program instituted by the Company, an
applicable exercise form provided by the Company (the "Exercise Form") duly
executed by the Warrantholder of record and specifying the number of Warrant
Shares to be purchased, to the Company at the office of the Company located at
3232 McKinney Avenue, Suite 1000, Dallas, Texas 75204 (or such other office or
agency of the Company as it may designate by notice to the Warrantholder at the
address of such Warrantholder appearing on the books of the Company) during
normal business hours on any day (a "Business Day") other than a Saturday,
Sunday or a day on which the New York Stock Exchange is authorized to close or
on which the Company is otherwise closed for business (a "Nonbusiness Day") on
or after 9:00 A.M. New York City time on the applicable Exercise Date but not
later than 5:00 P.M., New York City time, on the Expiration Date (or 5:00 P.M.,
New York City time, on the next succeeding Business Day, if the Expiration Date
is a Nonbusiness Day), (b) delivery of payment to the Company in cash or by
certified or official bank check in New York Clearing House Funds, of the
Exercise Price for the number of Warrant Shares specified in the Exercise Form
(such payment may be made by the Warrantholder directly or by a designated
broker pursuant to the broker-assisted cashless exercise program instituted by
the Company) and (c) such documentation as to the identity and authority of the
Warrantholder as the Company may reasonably request.  Such Warrant Shares shall
be deemed by the Company to be issued to the Warrantholder as the record holder
of such Warrant Shares as of the close of business on the date on which this
Warrant shall have been surrendered and payment made for the Warrant Shares as
aforesaid. Certificates for the Warrant Shares specified in the Exercise Form
shall be delivered to the Warrantholder (or designated broker, as the case may
be) as promptly as practicable, and in any event within 10 business days,
thereafter.  The stock certificates so delivered shall be in denominations of at
least 1,000 shares each or such other denomination as may be specified by the
Warrantholder and agreed upon by

                                       3
<PAGE>

the Company, and shall be issued in the name of the Warrantholder or such other
name as shall be designated in the Exercise Form. If this Warrant shall have
been exercised only in part, the Company shall, at the time of delivery of the
certificates for the Warrant Shares, deliver to the Warrantholder (or designated
broker, as the case may be) a new Warrant evidencing the rights to purchase the
remaining Warrant Shares, which new Warrant shall in all other respects be
identical with this Warrant. No adjustments or payments shall be made on or in
respect of Warrant Shares issuable on the exercise of this Warrant for any cash
dividends paid or payable to holders of record of Common Stock prior to the date
as of which the Warrantholder shall be deemed to be the record holder of such
Warrant Shares.

     (c)  With the consent of the Compensation Committee, and subject at all
times to, and only to the extent, if any, permitted under and in accordance
with, laws and regulations and other binding obligations or provisions
applicable to the Company, the Company may make a loan to the Warrantholder with
respect to the exercise of the Warrant, including the payment by the
Warrantholder of any or all federal, state and local income or other taxes due
in connection with any exercise.  The interest on such loan shall be the
Company's cost of money plus an additional 0.5% at the time the loan is made and
such loan shall be made with recourse against the Warrantholder. The
Compensation Committee shall have the full authority to determine any other
terms and provisions of such a loan.

     (d)  In the event the Company sells one or more of its subsidiaries or
businesses during fiscal 2001, the earnings targets set forth in Section 1.1(a)
would be reduced by the earnings budgeted for such divested businesses or
subsidiaries for the remainder of fiscal 2001 (including a reduction relating to
any associated write down of goodwill, if any) and increased by the actual
earnings attributable to any funds received as a result of such sale, all as
determined by the Compensation Committee.

     1.2    Vesting and Exercise.  Once vested, this Warrant may be exercised as
            --------------------
to such vested portion whether or not at the time of such exercise the
Warrantholder is an employee of (or consultant to) the Company (or one or more
of its subsidiaries); however, this Warrant shall automatically terminate as to
any unvested portion at any such time as the Warrantholder is no longer employed
by (or a consultant to) the Company (or any of its subsidiaries).  If this
Warrant is not exercised prior to 5:00 P.M. on the Expiration Date (or the next
succeeding Business Day, if the Expiration Date is a Nonbusiness Day), this
Warrant, or any new Warrant issued pursuant to Section 1.1, shall cease to be
exercisable and shall become void and all rights of the Warrantholder hereunder
shall cease.  This Warrant shall not be exercisable, and no Warrant Shares shall
be issued hereunder, prior to 9:00 A.M., New York City time, on the applicable
Exercise Date.

     1.3    Payment of Taxes.  The issuance of certificates for Warrant Shares
            ----------------
shall be made without charge to the Warrantholder for any stock transfer or
other issuance tax in respect thereto; provided, however, that the Warrantholder
                                       --------  -------
shall be required to pay any and all taxes which may be payable in respect to
any transfer involved in the issuance and delivery of any certificates for
Warrant Shares in a name other than that of the then Warrantholder as reflected
upon the books of the Company.

     1.4    Divisibility of Warrant.  This Warrant may be divided into warrants
            -----------------------
representing one Warrant Share or multiples thereof, upon surrender at the
principal office of the Company on

                                       4
<PAGE>

any Business Day, without charge to any Warrantholder, except as provided below.
The Warrantholder will be charged for reasonable out-of-pocket costs incurred by
the Company in connection with the division of this Warrant into Warrants
representing fewer than one thousand (1,000) Warrant Shares. Upon any such
division, the Warrants may be transferred of record to a name other than that of
the Warrantholder of record; provided, however, that the Warrantholder shall be
                             --------  -------
required to pay any and all transfer taxes with respect thereto.

     2.   Reservation and Listing of Shares, Etc.
          --------------------------------------

     All Warrant Shares which are issued upon the exercise of the rights
represented by this Warrant shall, upon issuance and payment of the Exercise
Price, be validly issued, fully paid and nonassessable and free from all taxes,
liens, security interests, charges and other encumbrances with respect to the
issue thereof other than taxes in respect of any transfer occurring
contemporaneously with such issue.  During the period within which this Warrant
may be exercised, the Company shall at all times have authorized and reserved,
and keep available free from preemptive rights, a sufficient number of shares of
Common Stock to provide for the exercise of this Warrant, and shall at its
expense use its best efforts to procure such listing thereof (subject to
official notice of issuance) as then may be required on all stock exchanges on
which the Common Stock is then listed or on the Nasdaq National Market.  The
Company shall, from time to time, take all such action as may be required to
assure that the par value per share of the Warrant Shares is at all times equal
to or less than the then effective Exercise Price.

     3.   Exchange, Loss or Destruction of Warrant.
          ----------------------------------------

     If permitted by Section 1.4 and in accordance with the provisions thereof,
upon surrender of this Warrant to the Company with a duly executed instrument of
assignment and funds sufficient to pay any transfer tax, the Company shall,
without charge, execute and deliver a new Warrant of like tenor in the name of
the assignee named in such instrument of assignment and this Warrant shall
promptly be canceled.  Upon receipt by the Company of evidence satisfactory to
it of the loss, theft, destruction or mutilation of this Warrant, and, in the
case of loss, theft or destruction, of such bond or indemnification as the
Company may reasonably require, and, in the case of such mutilation, upon
surrender and cancellation of this Warrant, the Company will execute and deliver
a new Warrant of like tenor.  The term "Warrant" as used herein includes any
Warrants issued in substitution or exchange of this Warrant.

     4.   Ownership of Warrant.
          --------------------

     The Company may deem and treat the person in whose name this Warrant is
registered as the holder and owner hereof (notwithstanding any notations of
ownership or writing hereon made by anyone other than the Company) for all
purposes and shall not be affected by any notice to the contrary, until
presentation of this Warrant for registration of transfer as provided in Section
1.1 or in Section 3.

     5.   Certain Adjustments.
          -------------------

     The Exercise Price at which Warrant Shares may be purchased hereunder, and
the number of Warrant Shares to be purchased upon exercise hereof, are subject
to change or adjustment as follows:

                                       5
<PAGE>

     5.1  The number of Warrant Shares purchasable upon the exercise of this
Warrant and the Exercise Price shall be subject to adjustment as follows:

     (a)  In case the Company shall (i) pay a dividend in shares of Common Stock
or make a distribution in shares of Common Stock, (ii) subdivide its outstanding
shares of Common Stock into a greater number of shares of Common Stock, (iii)
combine its outstanding shares of Common Stock into a smaller number of shares
of Common Stock or (iv) issue by reclassification of its shares of Common Stock
other securities of the Company (including any such reclassification in
connection with a consolidation or merger in which the Company is the surviving
corporation), the number of Warrant Shares purchasable upon exercise of this
Warrant (and the Exercise Price) shall be adjusted so that the Warrantholder
shall be entitled to receive (for the aggregate Exercise Price) the kind and
number of Warrant Shares or other securities of the Company that he would have
owned or have been entitled to receive after the happening of any of the events
described above, had this Warrant been exercised immediately prior to the
happening of such event or any record date with respect thereto.  An adjustment
made pursuant to this paragraph (a) shall become effective immediately after the
effective date of such event retroactive to the record date, if any, for such
event.

     (b)  In case the Company shall:

          (i)  issue rights, options or warrants to all holders of its
          outstanding Common Stock, without any charge to such holders,
          entitling them to subscribe for or purchase shares of Common Stock at
          a price per share which is lower at the record date for the
          determination of stockholders entitled to receive such rights, options
          or warrants than the then current market price per share of Common
          Stock, or

          (ii) distribute to all holders of its shares of Common Stock evidences
          of its indebtedness or assets (excluding cash dividends or
          distributions payable out of consolidated earnings or earned surplus
          and dividends or distributions referred to in paragraph (a) of this
          Section 5.1) or rights, options or warrants, or convertible or
          exchangeable securities, containing the right to subscribe for or
          purchase shares of Common Stock,

appropriate adjustments shall be made to the number of Warrant Shares
purchasable upon the exercise of the Warrant and/or the Exercise Price in order
to preserve the relative rights and interests of the Warrantholders, such
adjustments to be made by the good faith determination of the Board of Directors
of the Company.

     5.2  Voluntary Adjustment by the Company.  The Company may, at its option,
          -----------------------------------
at any time during the term of the Warrants, reduce the then current Exercise
Price to any amount, consistent with applicable law, deemed appropriate by the
Board of Directors of the Company.

     5.3  Notice of Adjustment.  Whenever the number of Warrant Shares or the
          --------------------
Exercise Price of such Warrant Shares is adjusted, as herein provided, the
Company shall promptly mail first class, postage prepaid, to all Warrantholders,
notice of such adjustment.

                                       6
<PAGE>

     5.4  No Adjustment for Cash Dividends.  No adjustment in respect of any
          --------------------------------
cash dividends shall be made during the term of this Warrant or upon the
exercise of this Warrant.

     5.5  Preservation of Purchase Rights Upon Merger, Consolidation, etc.  In
          ---------------------------------------------------------------
case of any consolidation of the Company with or merger of the Company into
another corporation or in case of any sale, transfer or lease to another
corporation of all or substantially all of the property of the Company, the
Company or such successor or purchasing corporation, as the case may be, shall
execute with the Warrantholders an agreement that the Warrantholders shall have
the right thereafter upon payment of the Exercise Price in effect immediately
prior to such action to purchase upon exercise of this Warrant the kind and
amount of shares and other securities and property that such holder would have
owned or have been entitled to receive after the happening of such
consolidation, merger, sale, transfer or lease had this Warrant been exercised
immediately prior to such action; provided, however, that no adjustment in
                                  --------  -------
respect of cash dividends, interest or other income on or from such shares or
other securities and property shall be made during the term of this Warrant or
upon the exercise of this Warrant.  Such agreement shall provide for
adjustments, which shall be as nearly equivalent as practicable to the
adjustments provided for in this Section 5.  The provisions of this Section 5.5
shall apply similarly to successive consolidations, mergers, sales, transfers or
leases.

     6.   Registration Rights of Warrant Shares on Form S-8
          -------------------------------------------------

     On or prior to January 31, 2002, the Company shall file a registration
statement covering the Warrant Shares on a Form S-8, which registration
statement shall be effective upon the filing thereof. The Company shall use its
best efforts to keep such Form S-8 current and effective until the earlier of
the Expiration Date or the date this Warrant has been exercised in full.  The
Company shall use its best efforts to list the Warrant Shares on any securities
exchange (or on the Nasdaq National Market) on which other shares of Common
Stock are listed.

     7.   Miscellaneous.
          -------------

     7.1  Entire Agreement.  This Warrant constitutes the entire agreement
          ----------------
between the Company and the Warrantholder with respect to this Warrant and the
Warrant Shares.

     7.2  Binding Effects; Benefits.  This Warrant shall inure to the benefit of
          -------------------------
and shall be binding upon the Company, the Warrantholder and holders of Warrant
Shares and their respective heirs, legal representatives, successors and
assigns.  Nothing in this Warrant, expressed or implied, is intended to or shall
confer on any person other than the Company, the Warrantholders and holders of
Warrant Shares, or their respective heirs, legal representatives, successors or
assigns, any rights, remedies, obligations or liabilities under or by reason of
this Warrant or the Warrant Shares.

     7.3  Amendments and Waivers.  This Warrant may not be modified or amended
          ----------------------
except by an instrument in writing signed by the Company and Warrantholders that
hold Warrants entitling them to purchase at least 50% of the Warrant Shares.
The Company, any Warrantholder or holder of Warrant Shares may, by an instrument
in writing, waive compliance by the other party with any term or provision of
this Warrant on the part of such other party hereto to be performed or complied
with.  The waiver by any such party of a breach of any term or provision of this
Warrant shall not be construed as a waiver of any subsequent breach.

                                       7
<PAGE>

     7.4  Section and Other Headings.  The section and other headings contained
          --------------------------
in this Warrant are for reference purposes only and shall not be deemed to be a
part of this Warrant or to affect the meaning or interpretation of this Warrant.

     7.5  Further Assurances.  Each of the Company, the Warrantholders and
          ------------------
holders of Warrant Shares shall do and perform all such further acts and things
and execute and deliver all such other certificates, instruments and/or
documents (including without limitation, such proxies and/or powers of attorney
as may be necessary or appropriate) as any party hereto may, at any time and
from time to time, reasonably request in connection with the performance of any
of the provisions of this Warrant.

     7.6  Notices.  All demands, requests, notices and other communications
          -------
required or permitted to be given under this Warrant shall be in writing and
shall be deemed to have been duly given if delivered personally or sent by
United States certified or registered first class mail, postage prepaid, to the
parties hereto at the following addresses or at such other address as any party
hereto shall hereafter specify by notice to the other party hereto:

     (a)  if to the Company, addressed to:

               F.Y.I. Incorporated
               3232 McKinney Avenue
               Suite 1000
               Dallas, Texas 75204
               Attention:  President

     (b)  if to any Warrantholder or holder of Warrant Shares, addressed to the
address of such person appearing on the books of the Company.

Except as otherwise provided herein, all such demands, requests, notices and
other communications shall be deemed to have been received on the date of
personal delivery thereof or on the third Business Day after the mailing
thereof.

     7.7  Separability.  Any term or provision of this Warrant that is invalid
          ------------
or unenforceable in any jurisdiction shall, as to such jurisdiction, be
ineffective to the extent of such invalidity or unenforceability without
rendering invalid or unenforceable any other term or provision of this Warrant
or affecting the validity or enforceability of any of the terms or provisions of
this Warrant in any other jurisdiction.

     7.8  Fractional Shares.  No fractional shares or scrip representing
          -----------------
fractional shares shall be issued upon the exercise of this Warrant.  With
respect to any fraction of a share called for upon any exercise hereof, the
Company shall pay to the Warrantholder an amount in cash equal to such fraction
multiplied by the current market price (as determined as of the date of
exercise, and with reference to the applicable trading market, in accordance
with paragraph (d) of Section 5.1) of a share of such stock as of the date of
such exercise.

     7.9  Rights of the Holder.  The Warrantholder shall not, solely by virtue
          --------------------
of this Warrant, be entitled to any rights of a stockholder of the Company,
either at law or in equity.

                                       8
<PAGE>

     7.10  Governing Law.  This Warrant shall be deemed to be a contract made
           -------------
under the laws of the State of Delaware and for all purposes shall be governed
by and construed in accordance with the laws of such State applicable to
contracts made and performed in Delaware.

     7.11  Effect of Stock Splits, etc. Whenever any rights under this Agreement
           ---------------------------
are available only when at least a specified minimum number of Warrant Shares is
involved, such number shall be appropriately adjusted to reflect any stock
split, stock dividend, combination of securities into a smaller number of
securities or reclassification of stock.

                                       9
<PAGE>

     IN WITNESS WHEREOF, the Company has caused this Warrant to be signed by its
duly authorized officer.


                                        F.Y.I. INCORPORATED



                                        By:    /s/ Ed H. Bowman, Jr.
                                               ---------------------
                                        Name:  Ed H. Bowman, Jr.
                                        Title: President and
                                               Chief Executive Officer



Dated:  January 24, 2001

                                       10
<PAGE>

                                 EXERCISE FORM
                                 -------------

                 (To be executed upon exercise of this Warrant)


     The undersigned, the record holder of this Warrant, hereby irrevocably
elects to exercise the right, represented by this Warrant, to purchase
__________ of the Warrant Shares and herewith tenders payment for such Warrant
Shares to the order of F.Y.I. INCORPORATED, in the amount of $_______ in
accordance with the terms of this Warrant.  The undersigned requests that a
certificate for such Warrant Shares be registered in the name of
_________________________________ and that such certificate be delivered to
_________________________ whose address is______________________________________
_____________________________.


Date________________                         Signature__________________________

                                       11