Section 5.1 Section 5.1 Inverse Functions
Subsection 1. Use new vocabulary
Subsubsection Definitions
Write a definition or description for each term:
- Inverse function
- Inverse function notation
- Horizontal line test
- One-to-one function
- Symmetric about \(y=x\)
Subsubsection Exercise
Checkpoint 5.1.
Illustrate each term above for the following situation:
The sales tax \(T\) on an item that costs \(p\) dollars is given by the function \(T=f(p)=1.15p\)
- The inverse function gives the price \(p\) of an item whose sales tax is \(T\) dollars.
- \(\displaystyle p=f^{-1}(T)\)
- The graph of \(T=f(p)=1.15p\) is linear, and so passes the horizontal line test.
- A function that passes the horizontal line test is one-to-one: for each value of \(p\) there is only one value of \(T\text{,}\) and vice versa.
Subsection 2. Solve an equation for a variable
When finding a formual for an inverse function, we need to solve for one variable in terms of the other.
Subsubsection Examples
Example 5.2.
Solve \(~y=\sqrt{x^3-4}~\) for \(x\) in terms of \(y\text{.}\)
Example 5.3.
Solve \(~y=\dfrac{x+1}{x-2}~\) for \(x\) in terms of \(y\text{.}\)
Subsubsection Exercises
Checkpoint 5.4.
Solve \(~y=5-4\sqrt{x+2}~\) for \(x\) in terms of \(y\text{.}\)
Checkpoint 5.5.
Solve \(~y=\dfrac{3}{\sqrt[3]{x+6}}~\) for \(x\) in terms of \(y\text{.}\)
Checkpoint 5.6.
Solve \(~y=(2x-3)^3+1~\) for \(x\) in terms of \(y\text{.}\)
Checkpoint 5.7.
Solve \(~y=\dfrac{3x-1}{2x+1}~\) for \(x\) in terms of \(y\text{.}\)
Subsection 3. Use function notation
Keep in mind that the notation for an inverse function, \(f^{-1}(x)\text{,}\) does not mean the reciprocal of the function. This is a different use of the notation from how it is used as an exponent.
Subsubsection Examples
Example 5.8.
\(f(x)=2x-3~\text{.}\) Find formulas for:
- \(\displaystyle g(x)=\dfrac{1}{f(x)}\)
- \(\displaystyle h(x)=-f(x)\)
- \(\displaystyle j(x)=f^{-1}(x)\)
- \(g(x)\) is the reciprocal of \(f(x)\text{:}\) \(g(x)=\dfrac{1}{2x-3}\)
- \(h(x)\) is the negative or opposite of \(f(x)\text{:}\) \(h(x)=3-2x\)
- To find the inverse function for \(f(x)\text{,}\) we write \(~y = 2x-3~\) and solve for \(x\text{.}\)\begin{align*} y \amp = 2x-3 \amp \amp \blert{\text{Add 3 to both sides.}}\\ y+3 \amp = 2x \amp \amp \blert{\text{Divide both sides by 2.}}\\ \dfrac{y+3}{2} \amp = x \end{align*}We can use any variables for a function, so we switch back to \(x\) for the input and \(y\) for the output: \(~y = \dfrac{x+3}{2}~\text{.}\) Thus, the inverse function is\begin{equation*} j(x)=\dfrac{x+3}{2} \end{equation*}
Example 5.9.
\(f(x)=\sqrt[3]{x-4}~\text{.}\) Find formulas for:
- \(\displaystyle g(x)=\dfrac{1}{f(x)}\)
- \(\displaystyle h(x)=-f(x)\)
- \(\displaystyle j(x)=f^{-1}(x)\)
- \(g(x)\) is the reciprocal of \(f(x)\text{:}\) \(~g(x)=\dfrac{1}{\sqrt[3]{x-4}}\)
- \(h(x)\) is the negative or opposite of \(f(x)\text{:}\) \(~h(x)=-\sqrt[3]{x-4}\)
- To find the inverse function for \(f(x)\text{,}\) we write \(~y = \sqrt[3]{x-4}~\) and solve for \(x\text{.}\)\begin{align*} y \amp = \sqrt[3]{x-4} \amp \amp \blert{\text{Cube both sides.}}\\ y^3 \amp = x-4 \amp \amp \blert{\text{Add 4 to both sides.}}\\ y^3+4 \amp = x \end{align*}We can use any variables for a function, so we switch back to \(x\) for the input and \(y\) for the output:\begin{equation*} j(x)=x^3+4 \end{equation*}
Subsubsection Exercise
Checkpoint 5.10.
\(f(x)=2-\dfrac{1}{2}x~\text{.}\) Find formulas for:
- \(\displaystyle g(x)=\dfrac{1}{f(x)}\)
- \(\displaystyle h(x)=-f(x)\)
- \(\displaystyle j(x)=f^{-1}(x)\)
- \(\displaystyle g(x)=\dfrac{2}{4-x}\)
- \(\displaystyle h(x)=\dfrac{1}{2}x-2\)
- \(\displaystyle j(x)=4-2x\)
Checkpoint 5.11.
\(f(x)=\dfrac{1}{x+2}~\text{.}\) Find formulas for:
- \(\displaystyle g(x)=\dfrac{1}{f(x)}\)
- \(\displaystyle h(x)=-f(x)\)
- \(\displaystyle j(x)=f^{-1}(x)\)
- \(\displaystyle g(x)=x+2\)
- \(\displaystyle h(x)=\dfrac{-1}{x+2}\)
- \(\displaystyle j(x)=\dfrac{1-2x}{x}\)
Checkpoint 5.12.
\(f(x)=x^{3/4}~\text{.}\) Find formulas for:
- \(\displaystyle g(x)=\dfrac{1}{f(x)}\)
- \(\displaystyle h(x)=-f(x)\)
- \(\displaystyle j(x)=f^{-1}(x)\)
- \(\displaystyle g(x)=x^{-3/4}\)
- \(\displaystyle h(x)=-x^{3/4}\)
- \(\displaystyle j(x)=x^{4/3}\)
Checkpoint 5.13.
\(f(x)=(x-1)^3+2~\text{.}\) Find formulas for:
- \(\displaystyle g(x)=\dfrac{1}{f(x)}\)
- \(\displaystyle h(x)=-f(x)\)
- \(\displaystyle j(x)=f^{-1}(x)\)
- \(\displaystyle g(x)=\dfrac{1}{(x-1)^3+2}\)
- \(\displaystyle h(x)=-(x-1)^3-2\)
- \(\displaystyle j(x)=1+\sqrt[3]{x-2}\)