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13 problems match “piecewise functions”. Clear

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1. Iterated piecewise function value ●○○○○ 10 ptsFor integers $n$, let $f(n) = \begin{cases} n^2 + 1 & \text{if } n \text{ is odd}, \\ n^2 - 3n + 1 & \text{if } n \text{ is even}. \end{cases}$ Find $f(f(f(f(f(f(2))))))$.
2. Continuity of a piecewise linear function ●○○○○ 10 ptsLet $f(n) = 4n+3$ if $n<a$, and $f(n) = 7n-12$ if $n\ge a$. Find $a$ so that the graph of $y=f(n)$ is continuous.
3. Piecewise function coefficients sum ●○○○○ 10 ptsLet $f(x) = \left\{ \begin{array}{cl} ax+3 & \text{ if }x>0, \\ ab & \text{ if }x=0, \\ bx+c & \text{ if }x<0. \end{array} \right.$ If $f(2)=5$, $f(0)=5$, and $f(-2)=-10$, and $a$, $b$, and $c$ are nonnegative integers, then find $a+b+c$.
4. Odd function value at a positive point ●○○○○ 10 ptsGiven that $f(x)$ is an odd function, and when $x < 0$, $f(x) = x^3 + x^2$, find $f(2)$.
5. Continuity constant for a piecewise function ●○○○○ 10 ptsLet $f(x)=\begin{cases} kx^{2} & \text{if } x\le 2 \\ x+k & \text{if } x>2\end{cases}$. Find the value of the constant $k$ that makes $f$ continuous. Round to two decimal places.
6. Continuity of a piecewise function ●●○○○ 20 ptsLet $f(x) = \begin{cases} 5x^2+2 & \text{if } x\le a, \\ 11x & \text{if } x>a. \end{cases}$ Find the smallest possible value of $a$ for which the graph of $y=f(x)$ is continuous (the graph can be drawn without lifting your pencil from the paper). Give your answer as a decimal.
7. Maximum of a piecewise function ●●○○○ 20 ptsDefine a new operation $\oplus$ as follows: when $a \geq b$, $a \oplus b = a$; when $a < b$, $a \oplus b = b^2$. Let $f(x) = (1 \oplus x)\cdot x - (2 \oplus x)$ for $x \in [-2, 2]$. Find the maximum value of $f(x)$.
8. Doubling map sequence term ●●○○○ 20 ptsA sequence $\{a_n\}$ satisfies $a_{n+1}=2a_n$ if $0 \leqslant a_n < \frac{1}{2}$ and $a_{n+1}=2a_n-1$ if $\frac{1}{2} \leqslant a_n < 1$, for every positive integer $n$, and $a_1=\frac{6}{7}$. Find $a_{24}$. Round to two decimal places.
9. Making a piecewise function differentiable at a join ●●○○○ 20 ptsLet $f(x)=ax+b$ for $x\le 2$ and $f(x)=2-x^2$ for $x>2$. Find $a$ and $b$ so that $f$ is differentiable at $x=2$.
10. Smallest bound on function differences ●●●○○ 50 ptsThe function $f(x)$ is defined on $[0,1]$ and satisfies $f(0)=f(1)$, and for any $x, y \in [0,1]$ with $x \neq y$, we have $|f(x)-f(y)|<|x-y|$. Find the smallest real number $m$ such that for every such function $f$ and any $x, y \in [0,1]$, we have $|f(x)-f(y)|<m$. Give your answer as a decimal.

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