Chapter 3

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2) .\) \(f(x)=\frac{x-2}{x+3}\)

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(m(x)=\frac{1}{2} x^{3}\)

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\). \(f(x)=5 x+7, \quad g(x)=4-2 x^{2}\)

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2) .\) \(f(x)=3^{x}\)

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(n(x)=\frac{1}{3}|x-2|\)

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\). \(f(x)=\sqrt{x+4}, \quad g(x)=12-x^{3}\)

For the following exercises, evaluate the expressions, given functions \(f, g,\) and \(h:\) \(f(x)=3 x-2\) \(g(x)=5-x^{2}\) \(h(x)=-2 x^{2}+3 x-1\) \(3 f(1)-4 g(-2)\)

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(p(x)=\left(\frac{1}{3} x\right)^{3}-3\)

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\). \(f(x)=\frac{1}{x+2}, \quad g(x)=4 x+3\)

For the following exercises, evaluate the expressions, given functions \(f, g,\) and \(h:\) \(f(x)=3 x-2\) \(g(x)=5-x^{2}\) \(h(x)=-2 x^{2}+3 x-1\) \(f\left(\frac{7}{3}\right)-h(-2)\)

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(q(x)=\left(\frac{1}{4} x\right)^{3}+1\)

For the following exercises, use the functions \(f(x)=2 x^{2}+1\) and \(g(x)=3 x+5\) to evaluate or find the composite function as indicated. \(f(g(2))\)

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(a(x)=\sqrt{-x+4}\)

For the following exercises, use the functions \(f(x)=2 x^{2}+1\) and \(g(x)=3 x+5\) to evaluate or find the composite function as indicated. \(f(g(x))\)

For the following exercises, graph \(y=x^{2}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. [-10,10]

For the following exercises, use the functions \(f(x)=2 x^{2}+1\) and \(g(x)=3 x+5\) to evaluate or find the composite function as indicated. \(g(f(-3))\)

For the following exercises, use the functions \(f(x)=2 x^{2}+1\) and \(g(x)=3 x+5\) to evaluate or find the composite function as indicated. \((g \circ g)(x)\)

For the following exercises, graph \(y=x^{3}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. [-0.1,0.1]

For the following exercises, use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). Find \((f \circ g)(x)\) and \((g \circ f)(x)\). Compare the two answers.

For the following exercises, use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). Find \((f \circ g)(2)\) and \((g \circ f)(2)\)

For the following exercises, use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). What is the domain of $$ (g \circ f)(x) ? $$

For the following exercises, graph \(y=\sqrt{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. \([0,0.01]\)

For the following exercises, use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). What is the domain of \((f \circ g)(x) ?\)

For the following exercises, graph \(y=\sqrt{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. \([0,100]\)

Let \(f(x)=\frac{1}{x}\). (a) Find \((f \circ f)(x)\). (b) Is \((f \circ f)(x)\) for any function \(f\) the same result as the answer to part (a) for any function? Explain.

For the following exercises, let \(F(x)=(x+1)^{5}, f(x)=x^{5},\) and \(g(x)=x+1\). True or False: \((g \circ f)(x)=F(x)\).

For the following exercises, graph \(y=\sqrt[3]{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. [-0.001,0.001]

For the following exercises, let \(F(x)=(x+1)^{5}, f(x)=x^{5},\) and \(g(x)=x+1\). True or False: \((f \circ g)(x)=F(x)\).

For the following exercises, graph \(y=\sqrt[3]{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. [-1000,1000]

For the following exercises, find the composition when \(f(x)=x^{2}+2\) for all \(x \geq 0\) and \(g(x)=\sqrt{x-2}\). \((f \circ g)(6) ; \quad(g \circ f)(6)\)

For the following exercises, graph \(y=\sqrt[3]{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. [-1,000,000,1,000,000]

For the following exercises, find the composition when \(f(x)=x^{2}+2\) for all \(x \geq 0\) and \(g(x)=\sqrt{x-2}\). \((g \circ f)(a) ; \quad(f \circ g)(a)\)

produced by a city with population \(p\) is given by \(G=f(p) \cdot G\) is measured in tons per week, and \(p\) is measured in thousands of people. (a) The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function \(f\). (b) Explain the meaning of the statement \(f(5)=2\)

For the following exercises, find the composition when \(f(x)=x^{2}+2\) for all \(x \geq 0\) and \(g(x)=\sqrt{x-2}\). \((f \circ g)(11) ; \quad(g \circ f)(11)\)

The function \(D(p)\) gives the number of items that will be demanded when the price is \(p\). The production \(\operatorname{cost} C(x)\) is the cost of producing \(x\) items. To determine the cost of production when the price is $$\$ 6$$ you would do which of the following? (a) Evaluate \(D(C(6))\). (b) Evaluate \(C(D(6))\). (c) Solve \(D(C(x))=6\). (d) Solve \(C(D(p))=6\).

Let \(f(t)\) be the number of ducks in a lake \(t\) years after 1990 . Explain the meaning of each statement: (a) \(f(5)=30\) (b) \(f(10)=40\)

The function \(A(d)\) gives the pain level on a scale of 0 to 10 experienced by a patient with \(d\) milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient's system after \(t\) minutes is modeled by \(m(t) .\) Which of the following would you do in order to determine when the patient will be at a pain level of \(4 ?\) (a) Evaluate \(A(m(4))\). (b) Evaluate \(m(A(4))\). c) Solve \(A(m(t))=4\). (d) Solve \(m(A(d))=4\).

Let \(h(t)\) be the height above ground, in feet, of a rocket \(t\) seconds after launching. Explain the meaning of each statement: (a) \(h(1)=200\) (b) \(h(2)=350\)

A store offers customers a \(30 \%\) discount on the price \(x\) of selected items. Then, the store takes off an additional \(15 \%\) at the cash register. Write a price function \(P(x)\) that computes the final price of the item in terms of the original price \(x\). (Hint: Use function composition to find your answer.)

Show that the function $$ f(x)=3(x-5)^{2}+7 \text { is not } $$ one-to-one.

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to \(r(t)=25 \sqrt{t+2},\) find the area of the ripple as a function of time. Find the area of the ripple at \(t=2\).

A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula \(r(t)=2 t+1,\) express the area burned as a function of time, \(t\) (minutes).

The radius \(r,\) in inches, of a spherical balloon is related: to the volume, \(V\), by \(r(V)=\sqrt[3]{\frac{3 V}{4 \pi}}\). Air is pumped into the balloon, so the volume after \(t\) seconds is given by \(V(t)=10+20 t\) (a) Find the composite function \(r(V(t))\). (b) Find the exact time when the radius reaches 10 inches.

The number of bacteria in a refrigerated food product is given by \(N(T)=23 T^{2}-56 T+1,3