Chapter 8

In Exercises \(1-4,\) state in words the inverse operation. Subtract 8

The graph of the function \(g(x)\) is a horizontal and/or vertical shift of the graph of \(f(x)=x^{3},\) shown in Figure \(8.19 .\) For each of the shifts described, sketch the graph of \(g(x)\) and find a formula for \(g(x)\). Shifted vertically up 3 units.

Find (a) The domain. (b) The range. $$ m(x)=9-x $$

Use substitution to compose the two functions. $$ y=u^{4} \text { and } u=x+1 $$

If we compose the two functions \(w=f(s)\) and \(q=\) \(g(w)\) using substitution, what is the input variable of the resulting function? What is the output variable?

State in words the inverse operation. Divide by 10 .

The graph of the function \(g(x)\) is a horizontal and/or vertical shift of the graph of \(f(x)=x^{3},\) shown in Figure \(8.19 .\) For each of the shifts described, sketch the graph of \(g(x)\) and find a formula for \(g(x)\). Shifted vertically down 2 units.

Find (a) The domain. (b) The range. $$ y=x^{2} $$

Use substitution to compose the two functions. $$ y=5 u^{3} \text { and } u=3-4 x $$

The graph of the function \(g(x)\) is a horizontal and/or vertical shift of the graph of \(f(x)=x^{3},\) shown in Figure \(8.19 .\) For each of the shifts described, sketch the graph of \(g(x)\) and find a formula for \(g(x)\). Shifted horizontally to the left 1 unit.

Find (a) The domain. (b) The range. $$ y=7 $$

Use substitution to compose the two functions. $$ w=r^{2}+5 \text { and } r=t^{3} $$

State in words the inverse operation. Take the ninth root.

The graph of the function \(g(x)\) is a horizontal and/or vertical shift of the graph of \(f(x)=x^{3},\) shown in Figure \(8.19 .\) For each of the shifts described, sketch the graph of \(g(x)\) and find a formula for \(g(x)\). Shifted horizontally to the right 2 units.

Find (a) The domain. (b) The range. $$ y=x^{2}-3 $$

Use substitution to compose the two functions. $$ p=2 q^{4} \text { and } D=5 p-1 $$

In Exercises \(5-8,\) find the sequence of operations to undo the sequence given. Multiply by 5 and then subtract \(2 .\)

The graph of the function \(g(x)\) is a horizontal and/or vertical shift of the graph of \(f(x)=x^{3},\) shown in Figure \(8.19 .\) For each of the shifts described, sketch the graph of \(g(x)\) and find a formula for \(g(x)\). Shifted vertically down 3 units and horizontally to the left 1 unit.

Find (a) The domain. (b) The range. $$ f(x)=x-3 $$

Use substitution to compose the two functions. $$ w=5 s^{3} \text { and } q=3+2 w $$

Find (a) The domain. (b) The range. $$ y=5 x-1 $$

Use substitution to compose the two functions. $$ P=3 q^{2}+1 \text { and } q=2 r^{3} $$

Find the sequence of operations to undo the sequence given. Raise to the \(5^{\text {th }}\) power and then multiply by 2 .

Find (a) The domain. (b) The range. $$ f(x)=\frac{1}{\sqrt{x-4}} $$

Use substitution to compose the two functions. $$ y=u^{2}+u+1 \text { and } u=x^{2} $$

Find (a) The domain. (b) The range. $$ y=\sqrt{x}+1 $$

Use substitution to compose the two functions. $$ y=2 u^{2}+5 u+7 \text { and } u=3 x^{3} $$

In Exercises \(9-12,\) show that composing the functions in either order gets us back to where we started. $$ y=7 x-5 \text { and } x=\frac{y+5}{7} $$

Find (a) The domain. (b) The range. $$ y=\sqrt{x+1} $$

Express the function as a composition of two simpler functions. $$ y=\sqrt{x^{2}+1} $$

Show that composing the functions in either order gets us back to where we started. $$ y=8 x^{3} \text { and } x=\sqrt[3]{\frac{y}{8}} $$

Find (a) The domain. (b) The range. $$ y=\frac{1}{x-2} $$

Express the function as a composition of two simpler functions. $$ y=5(x-2)^{3} $$

Show that composing the functions in either order gets us back to where we started. $$ y=x^{5}+1 \text { and } x=\sqrt[5]{y-1} $$

Find (a) The domain. (b) The range. $$ f(x)=\frac{1}{x+1}+3 $$

Express the function as a composition of two simpler functions. $$ y=3 x^{3}-2 $$

Show that composing the functions in either order gets us back to where we started. $$ y=\frac{10+x}{3} \text { and } x=3 y-10 $$

Find (a) The domain. (b) The range. $$ y=\sqrt{2 x-4} $$

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=\frac{(2 x+1)^{5}}{3} $$

In Exercises \(13-14\) (a) Write a function of \(x\) that performs the operations described. (b) Find the inverse and describe in words the sequence of operations in the inverse. Raise \(x\) to the fifth power, multiply by \(8,\) and then add \(4 .\)

Find (a) The domain. (b) The range. $$ h(z)=5+\sqrt{z-25} $$

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=\frac{17}{\left(1-x^{3}\right)^{4}} $$

The graphs are horizontal and/or vertical shifts of the graph of \(y=x^{2}\). Find a formula for each function graphed.

(a) Write a function of \(x\) that performs the operations described. (b) Find the inverse and describe in words the sequence of operations in the inverse. Subtract 5 , divide by 2 , and take the cube root.

A restaurant is open from \(2 \mathrm{pm}\) to 2 am each day, and a maximum of 200 clients can fit inside. If \(f(t)\) is the number of clients in the restaurant \(t\) hours after \(2 \mathrm{pm}\) each day, (a) What is reasonable domain for \(f ?\) (b) What is a reasonable range for \(f ?\)

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=\sqrt{5-x^{3}} $$

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=\frac{2}{\sqrt{1+\frac{1}{x}}} $$

A car's average gas mileage, \(G,\) is a function \(f(v)\) of the average speed driven, \(v\). What is a reasonable domain for \(f(v)\) ?

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=100(1+\sqrt{x})^{4} $$

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=\left(1+x+x^{2}\right)^{3} $$