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Precalculus for Team-Based Inquiry Learning PREVIEW Edition

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Section 2.6 Finding the Inverse Function (FN6)

Subsection 2.6.1 Activities

Remark 2.6.1.

A function is a process that converts a collection of inputs to a corresponding collection of outputs. One question we can ask is: for a particular function, can we reverse the process and think of the original function’s outputs as the inputs?
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Activity 2.6.2.

Temperature can be measured using many different units such as Fahrenheit, Celsius, and Kelvin. Fahrenheit is what is usually reported on the news each night in the United States, while Celsius is commonly used for scientific work. We will begin by converting between these two units. To convert from degrees Fahrenheit to Celsius use the following formula.
\begin{equation*} C=\dfrac{5}{9} (F-32) \end{equation*}
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(a)
Room temperature is around \(68\) degrees Fahrenheit. Use the above equation to convert this temperature to Celsius.
  1. \(\displaystyle 5.8\)
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  2. \(\displaystyle 20\)
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  3. \(\displaystyle 155.4\)
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  4. \(\displaystyle 293\)
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Answer.
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(b)
Solve the equation \(C=\dfrac{5}{9} (F-32)\) for \(F\) in terms of \(C\text{.}\)
  1. \(\displaystyle F=\dfrac{5}{9} C + 32\)
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  2. \(\displaystyle F=\dfrac{5}{9} C - 32\)
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  3. \(\displaystyle F=\dfrac{9}{5}( C + 32)\)
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  4. \(\displaystyle F=\dfrac{9}{5} C + 32\)
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Answer.
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(c)
Alternatively, \(20\) degrees Celsius is a fairly comfortable temperature. Use your solution for \(F\) in terms of \(C\) to convert this temperature to Fahrenheit.
  1. \(\displaystyle 43.1\)
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  2. \(\displaystyle -20.9\)
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  3. \(\displaystyle 93.6\)
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  4. \(\displaystyle 68\)
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Answer.
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Remark 2.6.3.

Notice that when you converted \(68\) degrees Fahrenheit, you got a value of \(20\) degrees Celsius. Alternatively, when you converted \(20\) degrees Celsius, you got \(68\) degrees Fahrenheit. This indicates that the equation you were given for \(C\) and the equation you found for \(F\) are inverses.
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Definition 2.6.4.

Let \(f\) be a function. If there exists a function \(g\) such that
\begin{equation*} f(g(x))=x \quad \text{and} \quad g(f(x))=x \end{equation*}
for all \(x\text{,}\) then we say \(f\) has an inverse function, or that \(g\) is the inverse of \(f\text{.}\) When a given function \(f\) has an inverse function, we usually denote it as \(f^{-1}\text{,}\) which is read as "\(f\) inverse".
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Remark 2.6.5.

An inverse is a function that "undoes" another function. For any input in the domain, the function \(g\) will reverse the process of \(f\text{.}\)
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Activity 2.6.6.

Let \(f(x)=2x+7\) and \(g(x)=\dfrac{x-7}{2}\text{.}\)
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(c)
What can you conclude about \(f(x)\) and \(g(x)\text{?}\)
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Answer.
\(f(x)\) and \(g(x)\) are inverses of each other.
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Activity 2.6.7.

It is important to note that in Definition 2.6.4 we say "if there exists a function," but we don’t guarantee that this is always the case. How can we determine whether a function has a corresponding inverse or not? Consider the following two functions \(f\) and \(g\) represented by the tables.
Table 2.6.8.
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\(x\) \(f(x)\)
\(0\) \(6\)
\(1\) \(4\)
\(2\) \(3\)
\(3\) \(4\)
\(4\) \(6\)
Table 2.6.9.
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\(x\) \(g(x)\)
\(0\) \(3\)
\(1\) \(1\)
\(2\) \(4\)
\(3\) \(2\)
\(4\) \(0\)
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(a)
Use the definition of \(g(x)\) in Table 2.6.9 to find an \(x\) such that \(g(x)=4\text{.}\)
  1. \(\displaystyle x=0\)
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  2. \(\displaystyle x=1\)
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  3. \(\displaystyle x=2\)
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  4. \(\displaystyle x=3\)
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  5. \(\displaystyle x=4\)
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Answer.
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(b)
Is it possible to reverse the input and output rows of the function \(g(x)\) and have the new table result in a function?
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Answer.
Yes. If you reverse the input and output values, you will see that every input goes to exactly one output (which defines a function).
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(c)
Use the definition of \(f(x)\) in Table 2.6.8 to find an \(x\) such that \(f(x)=4\text{.}\)
  1. \(\displaystyle x=0\)
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  2. \(\displaystyle x=1\)
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  3. \(\displaystyle x=2\)
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  4. \(\displaystyle x=3\)
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  5. \(\displaystyle x=4\)
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Answer.
B and D
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(d)
Is it possible to reverse the input and output rows of the function \(f(x)\) and have the new table result in a function?
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Answer.
If you reverse the input and output values, the resulting table is NOT a function. This is because the new input \(x=4\) will be mapping onto two \(y\)-values (\(1\) and \(3\)).
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Remark 2.6.10.

Some functions, like \(f(x)\) in Table 2.6.8, have a given output value that corresponds to two or more input values: \(f(0)=6\) and \(f(4)=6 \text{.}\) If we attempt to reverse the process of this function, we have a situation where the new input \(6\) would correspond to two potential outputs.
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Definition 2.6.11.

A one-to-one function is a function in which each output value corresponds to exactly one input.
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Remark 2.6.12.

A function must be one-to-one in order to have an inverse.
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Activity 2.6.13.

For each of the following graphs, determine if they represent a function that is one-to-one or not. If they are not one-to-one, what outputs have the same input?
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(a)
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
Answer.
This graph is not one-to-one
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(b)
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
Answer.
This graph is one-to-one
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(c)
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
Answer.
This graph is one-to-one
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(d)
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
Answer.
This graph is not one-to-one
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(e)
For each graph that was not one-to-one, draw a line connecting the points where two inputs had the same output. What do you notice about the lines?
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Answer.
A line between two points whose output is the same have the same \(y\)-value, so the line is horizontal.
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Observation 2.6.14.

When two outputs have the same input, this means that a horizontal line intersects the graph in two places. This leads us to the horizontal line test for one-to-one functions.
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Activity 2.6.16.

Consider the function \(f(x)=\dfrac{x-5}{3}\text{.}\)
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(a)
When you evaluate this expression for a given input value of \(x\text{,}\) what operations do you perform and in what order?
  1. divide by \(3\text{,}\) subtract \(5\)
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  2. subtract \(5\text{,}\) divide by \(3\)
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  3. add \(5\text{,}\) multiply by \(3\)
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  4. multiply by \(3\text{,}\) add \(5\)
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Answer.
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(b)
When you "undo" this expression to solve for a given output value of \(y\text{,}\) what operations do you perform and in what order?
  1. divide by \(3\text{,}\) subtract \(5\)
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  2. subtract \(5\text{,}\) divide by \(3\)
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  3. add \(5\text{,}\) multiply by \(3\)
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  4. multiply by \(3\text{,}\) add \(5\)
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Answer.
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(c)
This set of operations reverses the process for the original function, so can be considered the inverse function. Write an equation to express the inverse function \(f^{-1}\text{.}\)
  1. \(\displaystyle f^{-1}(x)=\dfrac{x}{3}-5\)
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  2. \(\displaystyle f^{-1}(x)=\dfrac{x-5}{3}\)
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  3. \(\displaystyle f^{-1}(x)=5(x+3)\)
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  4. \(\displaystyle f^{-1}(x)=3x+5\)
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Answer.
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(d)
Check your answer to the previous question by finding \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\text{.}\)
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Answer.
By completing this task, students should find that when they do the composition (in both directions), they will get an output of \(x\text{.}\)
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Observation 2.6.17.

To find the inverse of a one-to-one function, perform the reverse operations in the opposite order.
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Activity 2.6.18.

Let’s look at an alternate method for finding an inverse by solving the function for \(x\) and then interchanging the \(x\) and \(y\text{.}\)
\begin{equation*} h(x)=\dfrac{x}{x+1} \end{equation*}
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(a)
Interchange the variables \(x\) and \(y\text{.}\)
  1. \(\displaystyle y=\dfrac{x}{x+1}\)
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  2. \(\displaystyle x=\dfrac{y}{x+1}\)
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  3. \(\displaystyle x=\dfrac{y}{y+1}\)
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  4. \(\displaystyle x=\dfrac{x}{y+1}\)
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Answer.
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(b)
Eliminate the denominator.
  1. \(\displaystyle y(x+1)=x\)
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  2. \(\displaystyle x(x+1)=y\)
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  3. \(\displaystyle x(y+1)=x\)
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  4. \(\displaystyle x(y+1)=y\)
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Answer.
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(c)
Distribute and gather the \(y\) terms together.
  1. \(\displaystyle yx+y=x\)
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  2. \(\displaystyle x^{2}+x=y\)
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  3. \(\displaystyle xy-y=-x\)
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  4. \(\displaystyle xy=0\)
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Answer.
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(d)
Write the inverse function, by factoring and solving for \(y\text{.}\)
  1. \(\displaystyle h^{-1}(x)= \dfrac{x}{x-1}\)
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  2. \(\displaystyle h^{-1}(x)=\dfrac{x}{1-x}\)
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  3. \(\displaystyle h^{-1}(x)= \dfrac{-x}{1-x}\)
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  4. \(\displaystyle h^{-1}(x)= \dfrac{x+1}{x}\)
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Answer.
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Activity 2.6.19.

Find the inverse of each function, using either method. Check your answer using function composition.
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(a)
\(g(x)=\dfrac{4x-1}{7} \)
  1. \(\displaystyle g^{-1}(x)=\dfrac{7x+1}{4} \)
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  2. \(\displaystyle g^{-1}(x)= \dfrac{7x}{4}+1\)
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  3. \(\displaystyle g^{-1}(x)= \dfrac{4x+1}{7}\)
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  4. \(\displaystyle g^{-1}(x)=\dfrac{7}{4x-1} \)
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Answer.
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(b)
\(f(x)=2x^3-3 \)
  1. \(\displaystyle f^{-1}(x)= \frac{1}{2}x^{\frac{1}{3}}+3 \)
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  2. \(\displaystyle f^{-1}(x)= \left(\frac{1}{2}x\right)^{\frac{1}{3}}+3 \)
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  3. \(\displaystyle f^{-1}(x)= \left(\frac{1}{2}x+3\right)^{\frac{1}{3}} \)
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  4. \(\displaystyle f^{-1}(x)= \frac{1}{2}(x+3)^{\frac{1}{3}} \)
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Answer.
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Activity 2.6.20.

Consider the functions \(f(x)=x^2\) and \(g(x)=\sqrt{x}\text{.}\)
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(a)
Compute each of the following
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  1. \(\displaystyle \left(\sqrt{9}\right)^2\)
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  2. \(\displaystyle \left(\sqrt{25}\right)^2\)
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  3. \(\displaystyle \left(\sqrt{17}\right)^2\)
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Answer.
  1. \(\displaystyle 9\)
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  2. \(\displaystyle 25\)
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  3. \(\displaystyle 17\)
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(b)
Compute each of the following
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  1. \(\displaystyle \sqrt{3^2}\)
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  2. \(\displaystyle \sqrt{11^2}\)
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  3. \(\displaystyle \sqrt{37^2}\)
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  4. \(\displaystyle \sqrt{(-4)^2}\)
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Answer.
  1. \(\displaystyle 3\)
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  2. \(\displaystyle 11\)
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  3. \(\displaystyle 37\)
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  4. \(\displaystyle 4\)
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(c)
Are the functions \(f(x)=x^2\) and \(g(x)=\sqrt{x}\) one-to-one?
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Hint.
Use the horizontal line test.
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Answer.
\(f(x)=x^2\) is not one-to-one but \(g(x)=\sqrt{x}\) is.
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(d)
For which values of \(x\) is it true that \(f(g(x))=x\text{?}\)
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Answer.
\(\left(\sqrt{x}\right)^2=x\) for all \(x\) in the interval \([0, \infty)\text{.}\)
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(e)
For which values of \(x\) is it true that \(g(f(x))=x\text{?}\)
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Answer.
\(\sqrt{x^2}=x\) for all \(x\) in the interval \([0, \infty)\text{.}\)
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Observation 2.6.21.

While \(f(x)=x^2\) is not a one-to-one function and thus cannot have its inverse, we can restrict the domain to find an invertible function. In this case, considering \(f_0(x)=x^2\) defined only on the interval \([0,\infty)\text{,}\) \(f_0(x)\) is a one-to-one function with inverse \(f_0^{-1}(x)=\sqrt{x}\text{.}\)
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Remark 2.6.22.

When finding inverses algebraically, it is tempting to write \(\sqrt{x^2}=x\text{,}\) but this only true for non-negative \(x\)-values. In general, \(\sqrt{x^2}=|x|\text{.}\)
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Subsection 2.6.2 Exercises

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