# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 4

    Warm-up-exercises

### Exercise

Establish, for each $n\in \mathbb {N}$, whether the function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$
is injective and/or surjective.

### Exercise

Show that there exists a bijection between $\mathbb {N}$ and $\mathbb {Z}$.

### Exercise

Give examples of mappings
    $\varphi ,\psi \colon \mathbb {N} \longrightarrow \mathbb {N} ,$
such that $\varphi$ is injective, but not surjective, and $\psi$ is surjective, but not injective.

### Exercise

Let $L$ and $M$ be sets and let
    $F\colon L\longrightarrow M$
be a function. Let
    $G\colon M\longrightarrow L$
be another function such that $F\circ G=\operatorname {Id} _{M}\,$ and $G\circ F=\operatorname {Id} _{L}\,$. Show that $G$ is the inverse of $F$.

### Exercise

Determine the
composite functions $\varphi \circ \psi$ and $\psi \circ \varphi$ for the functions $\varphi ,\psi \colon \mathbb {R} \rightarrow \mathbb {R}$, defined by
    $\varphi (x)=x^{4}+3x^{2}-2x+5\,\,{\text{ and }}\,\,\psi (x)=2x^{3}-x^{2}+6x-1.$

### Exercise *

Let $L,M,N$ and $P$ be sets and let
    $F\colon L\longrightarrow M,x\longmapsto F(x),$
    $G\colon M\longrightarrow N,y\longmapsto G(y),$
and
    $H\colon N\longrightarrow P,z\longmapsto H(z),$
be functions. Show that
    $H\circ (G\circ F)=(H\circ G)\circ F\,.$

### Exercise *

Let $L,M,N$ be sets and let
    $f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N$
be functions with their composition
    $g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)).$
Show that if $g\circ f$ is
injective, then also $f$ is injective.

### Exercise

Let
    $f_{1},\ldots ,f_{n}\colon \mathbb {R} \longrightarrow \mathbb {R}$
be functions, which are increasing or decreasing, and let $f=f_{n}\circ \cdots \circ f_{1}$ be their
composition. Let $k$ be the number of the decreasing functions among the $f_{i}$'s. Show that if $k$ is even, then $f$ is
increasing, and if $k$ is odd, then $f$ is decreasing.

### Exercise

Calculate in the polynomial ring $\mathbb {C} [X]$ the product
    $((4+{\mathrm {i} })X^{2}-3X+9{\mathrm {i} })\cdot ((-3+7{\mathrm {i} })X^{2}+(2+2{\mathrm {i} })X-1+6{\mathrm {i} }).$

### Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Prove the following properties concerning the degree of a polynomial:
    $\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,,$
    $\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)\,.$

### Exercise

Show that in a polynomial ring over a field $K$, the following statement holds: if $P,Q\in K[X]$ are not zero, then also $PQ\neq 0$.

### Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Let $a\in K$. Prove that the evaluating function
    $\psi \colon K[X]\longrightarrow K,P\longmapsto P(a),$
satisfies the following properties (here let $P,Q\in K[X]$).
    $(P+Q)(a)=P(a)+Q(a)\,.$
    $(P\cdot Q)(a)=P(a)\cdot Q(a)\,.$
    $1(a)=1\,.$

### Exercise

Evaluate the polynomial
    $2X^{3}-5X^{2}-4X+7$
replacing the variable $X$ by the complex number $2-5{\mathrm {i} }$.

### Exercise

Perform, in the polynomial ring $\mathbb {Q} [X]$, the division with remainder ${\frac {P}{T}}$, where $P=3X^{4}+7X^{2}-2X+5$, and $T=2X^{2}+3X-1$.

### Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Show that every polynomial $P\in K[X]$, $P\neq 0$, can be decomposed as a product
    $P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q\,$
where $\mu _{j}\geq 1$ and $Q$ is a polynomial with no roots (no zeroes). Moreover, the different numbers $\lambda _{1},\ldots ,\lambda _{k}$ and the exponents $\mu _{1},\ldots ,\mu _{k}$ are uniquely determined apart from the order.

### Exercise

Let $F\in \mathbb {C} [X]$ be a non-constant polynomial. Prove that $F$ can be decomposed as a product of linear factors.

### Exercise

Determine the smallest real number for which the Bernoulli inequality with exponent $n=3$ holds.

### Exercise

Sketch the graph of the following rational functions
    $f=g/h\colon U\longrightarrow \mathbb {R} ,$
where each time $U$ is the complement set of the set of the zeros of the denominator polynomial $h$.
* $1/x$,
* $1/x^{2}$,
* $1/(x^{2}+1)$,
* $x/(x^{2}+1)$,
* $x^{2}/(x^{2}+1)$,
* $x^{3}/(x^{2}+1)$,
* $(x-2)(x+2)(x+4)/(x-1)x(x+1)$.

### Exercise

Let $P\in \mathbb {R} [X]$ be a polynomial with real coefficients and let $z\in \mathbb {C}$ be a root of $P$. Show that also the complex conjugate ${\overline {z}}$ is a root of $P$.
    Hand-in-exercises

### Exercise (3 marks)

Consider the set $M=\{1,2,3,4,5,6,7,8\}$ and the function
    $\varphi \colon M\longrightarrow M,x\longmapsto \varphi (x),$
defined by the following table
Compute $\varphi ^{1003}$, that is the $1003$-rd composition (or iteration) of $\varphi$ with itself.

### Exercise (2 marks)

Prove that a strictly increasing function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R}$
is injective.

### Exercise (3 marks)

Let $L,M,N$ be sets and let
    $f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N$
be functions with their composite
    $g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)).$
Show that if $g\circ f$ is
surjective, then also $g$ is surjective.

### Exercise (3 marks)

Compute in the polynomial ring $\mathbb {C} [X]$ the product
    $((4+{\mathrm {i} })X^{3}-{\mathrm {i} }X^{2}+2X+3+2{\mathrm {i} })\cdot ((2-{\mathrm {i} })X^{3}+(3-5{\mathrm {i} })X^{2}+(2+{\mathrm {i} })X+1+5{\mathrm {i} }).$

### Exercise (4 marks)

Perform, in the polynomial ring $\mathbb {C} [X]$ the division with remainder ${\frac {P}{T}}$, where
    $P=(5+X^{2}+{\mathrm {i} }X+3-{\mathrm {i} })X^{4}+X^{2}+{\mathrm {i} }X+3-{\mathrm {i} }X^{2}+(3-2X^{2}+{\mathrm {i} }X+3-{\mathrm {i} })X-1\,$
and
    $T=X^{2}+{\mathrm {i} }X+3-{\mathrm {i} }\,.$

### Exercise (4 marks)

Let $P\in \mathbb {R} [X]$ be a non-constant polynomial with real coefficients. Prove that $P$ can be written as a product of real polynomials of degrees $1$ or $2$.