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## Section2.8Additional Exercises

If $f\left( x\right) =\dfrac{1}{x-1}\text{,}$ then which of the following is equal to $f\left( \dfrac{1}{x}\right) ?$

1. $f(x)$

2. $-f(x)$

3. $xf(x)$

4. $-xf(x)$

5. $\dfrac{f(x)}{x}$

6. $-\dfrac{f(x)}{x}$

Answer

(d)

If $f(x)=\dfrac{x}{x+3}\text{,}$ then find and simplify $\dfrac{f(x)-f(2)}{x-2}\text{.}$

Answer

$3/\left[5(x+3)\right]$

If $f(x)=x^2\text{,}$ then find and simplify $\dfrac{f(3+h)-f(3)}{h}\text{.}$

Answer

$6+h$

What is the domain of

1. $f(x)=\dfrac{\sqrt{x-2}}{x^2-9}\text{?}$

2. $g(x)=\dfrac{\sqrt{x-2}}{x^2-9}\text{?}$

Answer
1. $[2,3)\cup(3,\infty)$

2. $(-\infty,-3)\cup(-3,3)\cup(3,\infty)$

Suppose that $f(x)=x^3$ and $g(x)=x\text{.}$ What is the domain of $\dfrac{f}{g}\text{?}$

Answer

$\left\{x:x\neq 0\right\}$

Suppose that $f(x)=3x-4\text{.}$ Find a function $g$ such that $(g\circ f)(x)=5x+2\text{.}$

Answer

$g(x)=(5x+26)/3$

Which of the following functions is one-to-one?

1. $f(x)=x^2+4x+3$

2. $g(x)=\vert x\vert+2$

3. $h(x)=\sqrt{x+1}$

4. $F(x)=\cos x\text{,}$ $-\pi\leq x\leq\pi$

5. $G(x)=e^x+e^{-x}$

Answer

(c)

What is the inverse of $f(x)=\ln\left(\dfrac{e^x}{e^x-1}\right)\text{?}$ What is the domain of $f^{-1}\text{?}$

Answer

$f^{-1}(x)=f(x)=\ln\left(\dfrac{e^x}{e^x-1}\right)$ and its domain is $(0,\infty)\text{.}$

Solve the following equations.

1. $e^{2-x}=3$

2. $e^{x^2}=e^{4x-3}$

3. $\ln\left(1+\sqrt{x}\right)=2$

4. $\ln(x^2-3)=\ln 2+\ln x$

Answer
1. $2-\ln 3$

2. 1,3

3. $(e^2-1)^2$

4. 3

Find the exact value of $\sin^{-1}\left(-\sqrt{2}/2\right)-\cos^{-1}\left(-\sqrt{2}/2\right)\text{.}$

Answer

$-\pi$

Find $\sin^{-1}\left(\sin(23\pi/5)\right)\text{.}$

Answer

$2\pi/5$

It can be proved that $f(x)=x^3+x+e^{x-1}$ is one-to-one. What is the value of $f^{-1}(3)\text{?}$

Answer

We notice that $f(1) = 1^3+1+e^{0} = 1+1+1=3\text{,}$ and so we must have that $f^{-1}(3) = 1\text{.}$

Sketch the graph of $f(x)=\left\{ \begin{array}{cc} -x \amp \text{ if } x\leq 0 \\ \tan ^{-1}x \amp \text{ if } x>0 \end{array} \right.$