Section 1
Inverse Functions
A study found that the trans fat content in fast-food products varied widely around the world, based on the type of frying oil used, as shown in the table. If the set of countries is the domain and the set of trans fat percentages is the range of the function consisting of the six pairs listed, is it a one-to-one function? Why or why not?
The table shows the number of uncontrolled hazardous waste sites in 2008 that require further investigation to determine whether remedies are needed under the Superfund program. The eight states listed are ranked in the top ten on the EPA's National Priority List.If this correspondence is considered to be a function that pairs each state with its number of uncontrolled waste sites, is it one-to-one? If not, explain why.
The road mileage between Denver, Colorado, and several selected U.S. cities is shown in the table. If we consider this as a function that pairs each city with a distance, is it a one-to-one function? How could we change the answer to this question by adding 1 mile to one of the distances shown?
Suppose you consider the set of ordered pairs $(x, y)$ such that $x$ represents a person in your mathematics class and $y$ represents that person's father. Explain how this function might not be a one-to-one function.
If a function is made up of ordered pairs in such a way that the same $y$-value appears in a correspondence with two different $x$-values, thenA. the function is one-to-oneB. the function is not one-to-oneC. its graph does not pass the vertical line testD. it has an inverse function associated with it.
Which equation defines a one-to-one function? Explain why the others are not, using specific examples.A. $f(x)=x$B. $f(x)=x^{2}$ C. $f(x)=|x|$D. $f(x)=-x^{2}+2 x-1$
Only one of the graphs illustrates a one-to-one function. Which one is it?(Check your book for graph)
If a function $f$ is one-to-one and the point $(p, q)$ lies on the graph of $f$ then which point must lie on the graph of $f^{-1} ?$A. $(-p, q)$B. $(-q,-p)$ C. $(p,-q)$D. $(q, p)$
If the function is one-to-one, find its inverse.$\{(3,6),(2,10),(5,12)\}$
If the function is one-to-one, find its inverse.$\left\{(-1,3),(0,5),(5,0),\left(7,-\frac{1}{2}\right)\right\}$
If the function is one-to-one, find its inverse.$\{(-1,3),(2,7),(4,3),(5,8)\}$
If the function is one-to-one, find its inverse.$\{(-8,6),(-4,3),(0,6),(5,10)\}$
If the function is one-to-one, find its inverse.$f(x)=2 x+4$
If the function is one-to-one, find its inverse.$f(x)=3 x+1$
If the function is one-to-one, find its inverse.$g(x)=\sqrt{x-3}, \quad x \geq 3$
If the function is one-to-one, find its inverse.$g(x)=\sqrt{x+2}, \quad x \geq-2$
If the function is one-to-one, find its inverse.$f(x)=3 x^{2}+2$
If the function is one-to-one, find its inverse.$f(x)=4 x^{2}-1$
If the function is one-to-one, find its inverse.$f(x)=x^{3}-4$
If the function is one-to-one, find its inverse.$f(x)=x^{3}+5$
Let $f(x)=2^{x} .$ We will see in the next section that this function is one-toone. Find each value, always working part (a) before part $(b)$.
(a) $f(3)$(b) $f^{-1}(8)$
(a) $f(4)$(b) $f^{-1}(16)$
(a) $f(0)$(b) $f^{-1}(1)$
(a) $f(-2)$(b) $f^{-1}\left(\frac{1}{4}\right)$
(a) Use the horizontal line test to determine whether the function is one-to-one. (b) If the function is one-to-one, then graph the inverse of the function. (Remember that if $f$ is one-to-one and $(a, b)$ is on the graph of $f,$ then $(b, a)$ is on the graph of $f^{-1} .$)(Check your book for graph)
Graph the function as a solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or curve).$f(x)=2 x-1$
Graph the function as a solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or curve).$f(x)=2 x+3$
Graph the function as a solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or curve).$g(x)=-4 x$
Graph the function as a solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or curve).$g(x)=-2 x$
Complete the table so that graphing the function will be easier.$f(x)=\sqrt{x}$$x \geq 0$$\begin{array}{c|c}{x} & {f(x)} \\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 4 & {}\end{array}$
Complete the table so that graphing the function will be easier.$f(x)=-\sqrt{x}$$x \geq 0$$\begin{array}{c|c}{x} & {f(x)} \\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 4 & {}\end{array}$
Complete the table so that graphing the function will be easier.$f(x)=x^{3}-2$$\begin{array}{c|c}{x} & {f(x)} \\ \hline-1 & {} \\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 2\end{array}$
Complete the table so that graphing the function will be easier.$f(x)=x^{3}+3$$\begin{array}{c|c}{x} & {f(x)} \\ \hline-2 & {} \\ \hline-1 & {} \\ \hline 0 & {} \\ \hline 1 & {}\end{array}$
Suppose that you are an agent for a detective agency. Today's function for your code is defined by $f(x)=4 x-5 .$ Find the rule for $f^{-1}$ algebraically.
You receive the following coded message today. (Read across from left to right.)$\begin{array}{lllllllllllll}{47} & {95} & {23} & {67} & {-1} & {59} & {27} & {31} & {51} & {23} & {7-1} & {43} & {779} & {43} & {-1} & {75} & {55} & {67}\end{array}$$\begin{array}{llllllllllllll}{31} & {71} & {75} & {27} & {15} & {23} & {67} & {15} & {-1} & {75} & {15} & {71} & {75} & {75} & {27} & {31} & {51}\end{array}$$\begin{array}{llllllllllllll}{23} & {71} & {31} & {51} & {715} & {71} & {43} & {31} & {7} & {15} & {11} & {367} & {15} & {-1} & {11}\end{array}$Use the letter/number assignment described earlier to decode the message.
Why is a one-to-one function essential in this encoding/decoding process?
Use $f(x)=x^{3}+4$ to encode your name, using the letter/number assignment described earlier.
Function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window.$f(x)=2 x-7$
Function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window.$f(x)=-3 x+2$
Function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window.$f(x)=x^{3}+5$
Function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window.$f(x)=\sqrt[3]{x+2}$
If $f(x)=4^{x},$ find value indicated. Use a calculator, and give the answer to the nearest hundredth.$f(3)$
If $f(x)=4^{x},$ find value indicated. Use a calculator, and give the answer to the nearest hundredth.$f\left(\frac{1}{2}\right)$
If $f(x)=4^{x},$ find value indicated. Use a calculator, and give the answer to the nearest hundredth.$f\left(-\frac{1}{2}\right)$
If $f(x)=4^{x},$ find value indicated. Use a calculator, and give the answer to the nearest hundredth.$f(2.73)$