problem stringclasses 3
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values | difficulty float64 3 5 | type stringclasses 1
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Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | 5 | Algebra | First, ensure the function values match at the boundary points by setting the expressions equal at x = 2 and x = -2. Solve the equations obtained from these conditions to find the unknown constants, then add them together. |
Step 1: Use the hint’s advice by ensuring the function values match at the boundary points. Start with x = 2. Since x = 2 falls in the interval -2 ≤ x ≤ 2, we have
f(2) = 2 – 5 = -3.
For x > 2, the function is given by f(x) = ax + 3. The limit as x approaches 2 from the right is
lim (x→2⁺) f(x) = 2a + 3. ... |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | 5 | Algebra | Formulate an equation by letting the total number of members equal the spots in the initial array plus two, and also equal the spots in the adjusted formation. Expand and simplify the equation to express one variable in terms of the other. Substitute back to express the total members in a single variable, then use the ... |
Step 1. (Using the hint’s first instruction)
Let the total number of band members be T. According to the problem, when the director arranges the band in a rectangular formation, there are m members per row and r rows but 2 members are left over. Thus, we can write
T = m·r + 2.
Also, after adjusting the formation by ... |
What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | 4 | 3 | Algebra | Break the polynomial into its individual terms, combine like terms, and then look for the term with the highest power of x that has a non-zero coefficient. |
Step 1: Break the polynomial into its individual terms (using the first part of the hint). The given polynomial is:
4 + 5x³ + 100 + 2πx⁴ + √10 x⁴ + 9.
Step 2: Combine like terms (using the second part of the hint).
• The constant terms are: 4, 100, and 9. Their sum is: 4 + 100 + 9 = 113.
• The x³ term is: ... |
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