Chapter 3

Use algebra to find the inverse of the given one-to-one function. $$f(x)=\sqrt[5]{\frac{3 x-1}{x-2}}$$

Compute and simplify the difference quotient of the function. $$f(t)=160,000-8000 t+t^{2}$$

Describe a sequence of transformations that will transform the graph of the function \(f\) into the graph of the function \(g.\) $$f(x)=x^{2}+x ; \quad g(x)=(x-3)^{2}+(x-3)+2$$

Find the rule of the function \(f \circ g,\) the domain of \(f \circ g,\) the rule of \(g \cdot f,\) and the domain of \(g \circ f\) $$f(x)=1 / x, \quad g(x)=\sqrt{x}$$

(a) Use the fact that the absolute value function is piecewise-defined (see Example 7) to write the rule of the given function as a piecewise-defined function whose rule does not include any absolute value bars. (b) Graph the function. $$g(x)=|x|-4$$

Each equation defines y as a function of \(x .\) Create a table that shows the values of the function for the given values of \(x\) $$y=x^{10}-100 x \quad x=-1,0,1,2,3,4$$

Use the Round-Trip Theorem on page 223 to show that \(g\) is the inverse of \(f\) $$f(x)=x+1, \quad g(x)=x-1$$

Compute and simplify the difference quotient of the function. $$V(x)=x^{3}$$

Describe a sequence of transformations that will transform the graph of the function \(f\) into the graph of the function \(g.\) $$f(x)=x^{2}+5 ; \quad g(x)=(x+2)^{2}+10$$

Find the rule of the function \(f \circ g,\) the domain of \(f \circ g,\) the rule of \(g \cdot f,\) and the domain of \(g \circ f\) $$f(x)=\frac{1}{2 x+1}, \quad g(x)=x^{2}-1$$

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$f(g(3))$$

Use the Round-Trip Theorem on page 223 to show that \(g\) is the inverse of \(f\) $$f(x)=2 x-6, \quad g(x)=\frac{x}{2}+3$$

Describe a sequence of transformations that will transform the graph of the function \(f\) into the graph of the function \(g.\) $$f(x)=\sqrt{x^{3}+5} ; \quad g(x)=-\frac{1}{2} \sqrt{x^{3}+5}-6$$

Compute and simplify the difference quotient of the function. $$A(r)=\pi r^{2}$$

(a) Use the fact that the absolute value function is piecewise-defined (see Example 7) to write the rule of the given function as a piecewise-defined function whose rule does not include any absolute value bars. (b) Graph the function. $$g(x)=|x+3|$$

Describe a sequence of transformations that will transform the graph of the function \(f\) into the graph of the function \(g.\) $$f(x)=\sqrt{x^{4}+x^{2}+1} ; g(x)=10-\sqrt{4 x^{4}+4 x^{2}+4}$$

Find the rules of the functions ff and \(f \circ f\) $$f(x)=x^{3}$$

Show that the function \(f(x)=|x|+|x-2|\) is constant on the interval \([0,2] .\) [ Hint: Use the definition of absolute value (see Example \(7 \text { ) to compute } f(x) \text { when } 0 \leq x \leq 2 .]\)

Assume \(h \neq 0 .\) Compute and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$ $$f(x)=x+1$$

Use the Round-Trip Theorem on page 223 to show that \(g\) is the inverse of \(f\) $$f(x)=\frac{-3}{2 x+5}, \quad g(x)=\frac{-3-5 x}{2 x}$$

Write the rule of a function g whose graph can be obtained from the graph of the function \(f\) by performing the transformations in the order given. \(f(x)=x^{2}+2 ;\) shift the graph horizontally 5 units to the left and then vertically upward 4 units.

Find the rules of the functions ff and \(f \circ f\) $$f(x)=(x-1)^{2}$$

Find the approximate location of all local maxima and minima of the function. $$f(x)=x^{3}-x$$

Assume \(h \neq 0 .\) Compute and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$ $$f(x)=-10 x$$

Use the Round-Trip Theorem on page 223 to show that \(g\) is the inverse of \(f\) $$f(x)=x^{5}, \quad g(x)=\sqrt[5]{x}$$

Write the rule of a function g whose graph can be obtained from the graph of the function \(f\) by performing the transformations in the order given. \(f(x)=x^{2}-x+1 ;\) reflect the graph in the \(x\) -axis, then shift it vertically upward 3 units.

Find the rules of the functions ff and \(f \circ f\) $$f(x)=1 / x$$

Find the approximate location of all local maxima and minima of the function. $$g(t)=-\sqrt{16-t^{2}}$$

Assume \(h \neq 0 .\) Compute and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$ $$f(x)=3 x+7$$

The amount of postage required to mail a first-class letter is determined by its weight. In this situation, is weight a function of postage? Or vice versa? Or both?

Use the Round-Trip Theorem on page 223 to show that \(g\) is the inverse of \(f\) $$f(x)=x^{3}-1, \quad g(x)=\sqrt[3]{x+1}$$

Write the rule of a function g whose graph can be obtained from the graph of the function \(f\) by performing the transformations in the order given. \(f(x)=\sqrt{x} ;\) shift the graph horizontally 6 units to the right, stretch it away from the \(x\) -axis by a factor of \(2,\) and shift it vertically downward 3 units.

Find the approximate location of all local maxima and minima of the function. $$h(x)=\frac{x}{x^{2}+1}$$

Assume \(h \neq 0 .\) Compute and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$ $$f(x)=x^{2}$$

Chinese philosopher Laotze (600 BC) said, "the farther one travels, the less one knows." Let \(x\) be the distance one travels, and \(y\) be the amount one knows. If Laotze is right, is \(y\) a function of \(x ?\) Is \(x\) a function of \(y ?\) Why or Why not?

Show that the inverse function of the function \(f\) whose rule is \(f(x)=\frac{2 x+1}{3 x-2}\) is \(f\) itself.

Write the rule of a function g whose graph can be obtained from the graph of the function \(f\) by performing the transformations in the order given. \(f(x)=\sqrt{-x} ;\) shift the graph horizontally 3 units to the left, then reflect it in the \(x\) -axis, and shrink it toward the \(x\) -axis by a factor of \(1 / 2\).

Verify that \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\) for every \(x\) $$f(x)=9 x+8, \quad g(x)=\frac{x-8}{9}$$

Find the approximate location of all local maxima and minima of the function. $$k(x)=x^{3}-3 x+1$$

Assume \(h \neq 0 .\) Compute and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$ $$f(x)=x-x^{2}$$

Could the following statement ever be the rule of a function? For input \(x,\) the output is the number whose square is \(x\) Why or why not? If there is a function with this rule, what is its domain and range?

List three different functions (other than the ones in Example 6 and Exercise 29 ), each of which is its own inverse. [Many correct answers are possible.]

Let \(f(x)=x^{2}+3 x,\) and let \(g(x)=f(x)+2\) (a) Write the rule of \(g(x)\) (b) Find the difference quotients of \(f(x)\) and \(g(x) .\) How are they related?

Verify that \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\) for every \(x\) $$f(x)=\sqrt[3]{x-1}, \quad g(x)=x^{3}+1$$

Find the approximate location of all local maxima and minima of the function. $$l(x)=\frac{1}{1+x^{2}}$$

Assume \(h \neq 0 .\) Compute and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$ $$f(x)=x^{3}$$

Let \(f(t)\) be the population of rabbits on Christy's property \(t\) years after she received 10 of them as a gift. $$\begin{array}{|c|c|}\hline t & f(t) \\\\\hline 0 & 10 \\\\\hline 1 & 23 \\\\\hline 2 & 48 \\\\\hline 3 & 64 \\\\\hline 4 & 70 \\\\\hline 5 & 71 \\\\\hline\end{array}$$ Compute the following, including units, or write "not enough information to tell." \(f^{-1}\) denotes the inverse function of \(f\). (a) \(f(2)\) (b) \(f^{-1}(48)\) (c) \(f^{-1}(71)\) (d) \(3 \cdot f^{-1}(70)\) (e) \(f^{-1}(2 \cdot 48)\) (f) \(f(70)\) (g) \(f^{-1}(4)\)

Let \(f(x)=x^{2}+5,\) and let \(g(x)=f(x-1)\) (a) Write the rule of \(g(x)\) and simplify. (b) Find the difference quotients of \(f(x)\) and \(g(x)\) (c) Let \(d(x)\) denote the difference quotient of \(f(x) .\) Show that the difference quotient of \(g(x)\) is \(d(x-1)\)

Verify that \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\) for every \(x\) $$f(x)=\sqrt[3]{x}+2, \quad g(x)=(x-2)^{3}$$

Assume \(h \neq 0 .\) Compute and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$ $$f(x)=\sqrt{x}$$