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# Week 4 Problem Set B.
Due: Thursday 9/28
Keep going ! Do the problems ! You go this !
### Reading.
Continue reading either (or both):
- Thomas' Calculus (12E) chapters 1.1, 1.2, and 1.3
- Stewart's Calculus (9E, with Clegg and Watson) chapters 1.1, 1.2, and 1.3
### Composition of functions.
1. If $f(x) = x - 1$ and $\displaystyle g(x) = \frac{1}{x+1}$, find each of the following:
1. $f(g(1/ 2))$
2. $g( f (1 / 2))$
3. $f(g(x))$
4. $g(f(x))$
2. Write a formula for $f\circ g \circ h$ if $f(x)=\sqrt{x+1}$, $\displaystyle g(x)= \frac{1}{x+4}$, $\displaystyle h(x)=\frac{1}{x}$.
3. Let $f(x) = x-3$, $g(x) = \sqrt{x}$, $h(x)=x^{3}$, and $j(x)=2x$. Express each of the following as a composition of functions involving one or more of $f,g,h,j$.
1. $y=2x-3$
2. $y=x^{3 /2}$
3. $y = x^{9}$
4. $y = x-6$
5. $y=2\sqrt{x-3}$
6. $y=\sqrt{x^{3}-3}$
4. Copy and complete each ? in the following table. $$
\begin{array}{} \\
g(x) & f(x) & (f\circ g)(x) \\ \hline
x-7 & \sqrt{x} & ? \\
x+2 & 3x & ? \\
? & \sqrt{x-5} & \sqrt{x^{2}-5} \\
\displaystyle\frac{x}{x-1} &\displaystyle \frac{x}{x-1} & ? \\
? & \displaystyle1+ \frac{1}{x} & x \\
\displaystyle\frac{1}{x} & ? & x \\
? & \displaystyle\frac{x-1}{x} & \displaystyle \frac{x}{x+1} \\
? & \sqrt{x} & |x|
\end{array}
$$
### Shifting graphs.
1. Match the equation listed below 1. to 4. to the graphs shown the four positions below: ![[1 teaching/smc-fall-2023-math-7/week-4/---files/Pasted image 20230920120854.png]]
1. $y=(x-1)^{2}-4$
2. $y=(x-2)^{2}+2$
3. $y= (x+2)^{2}+2$
4. $y=(x+3)^{2}-2$
2. In below figure shows parabolas in four different positions. They are all shifts of the graph of $y=-x^{2}$. Use the fact that they are all shifts of $y=-x^{2}$ to write down an equation for each of the graphs:![[1 teaching/smc-fall-2023-math-7/week-4/---files/Pasted image 20230920121024.png]]
3. The following shows the graph of some function $f(x)$ with domain $[0,2]$ and range $[0,1]$. Find the domain and range of each of the following functions, and sketch their graphs:![[1 teaching/smc-fall-2023-math-7/week-4/---files/Pasted image 20230920124114.png]]
1. $f(x)+2$
2. $f(x)-1$
3. $2f(x)$
4. $-f(x)$
5. $f(x+2)$
6. $f(x-1$)
7. $f(-x)$
8. $-f(x+1)+1$
### Combining functions.
1. Assume that $f$ is an even function, and $g$ is an odd function, and both $f$ and $g$ are defined on the real line $\mathbb{R}$. Which of the following, where defined, are even? Which are odd?
1. $fg$
2. $\frac{f}{g}$
3. $\frac{g}{f}$
4. $f^{2}=ff$
5. $g^{2}=gg$
6. $f\circ g$
7. $g\circ f$
8. $f\circ f$
9. $g\circ g$
2. Can a function be both even and odd? Explain your answer.
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