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Precalculus for Team-Based Inquiry Learning: 2024 Development Edition — Instructor Version

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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?

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=\frac{5}{9} (F-32) \end{equation*}
(a)
Room temperature is around 68 degrees Fahrenheit. Use the above equation to convert this temperature to Celsius.
  1. \(\displaystyle 5.8\)
  2. \(\displaystyle 20\)
  3. \(\displaystyle 155.4\)
  4. \(\displaystyle 293\)
(b)
Solve the equation \(C=\frac{5}{9} (F-32)\) for \(F\) in terms of \(C\text{.}\)
  1. \(\displaystyle F=\frac{5}{9} C + 32\)
  2. \(\displaystyle F=\frac{5}{9} C - 32\)
  3. \(\displaystyle F=\frac{9}{5}( C + 32)\)
  4. \(\displaystyle F=\frac{9}{5} C + 32\)
(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. \(\displaystyle 43.1\)
  2. \(\displaystyle -20.9\)
  3. \(\displaystyle 93.6\)
  4. \(\displaystyle 68\)

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.

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".

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 \(f\text{.}\)

Activity 2.6.6.

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 2.6.7.
\(x\) \(f(x)\)
\(0\) \(6\)
\(1\) \(4\)
\(2\) \(3\)
\(3\) \(4\)
\(4\) \(6\)
Table 2.6.8.
\(x\) \(g(x)\)
\(0\) \(3\)
\(1\) \(1\)
\(2\) \(4\)
\(3\) \(2\)
\(4\) \(0\)
(a)
Use the definition of \(g(x)\) in Table 2.6.8 to find an \(x\) such that \(g(x)=4\text{.}\)
  1. \(\displaystyle x=0\)
  2. \(\displaystyle x=1\)
  3. \(\displaystyle x=2\)
  4. \(\displaystyle x=3\)
  5. \(\displaystyle x=4\)
(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?
(c)
Use the definition of \(f(x)\) in Table 2.6.7 to find an \(x\) such that \(f(x)=4\text{.}\)
  1. \(\displaystyle x=0\)
  2. \(\displaystyle x=1\)
  3. \(\displaystyle x=2\)
  4. \(\displaystyle x=3\)
  5. \(\displaystyle x=4\)
(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?

Remark 2.6.9.

Some functions, like \(f(x)\) in Table 2.6.7, 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.

Definition 2.6.10.

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

Remark 2.6.11.

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

Activity 2.6.12.

Consider the function \(f(x)=\dfrac{x-5}{3}\text{.}\)
(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, subtract 5
  2. subtract 5, divide by 3
  3. add 5, multiply by 3
  4. multiply by 3, add 5
(b)
When you "undo" this expression to solve for a given ouput value of \(y\text{,}\) what operations do you perform and in what order?
  1. divide by 3, subtract 5
  2. subtract 5, divide by 3
  3. add 5, multiply by 3
  4. multiply by 3, add 5
(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. \(\displaystyle f^{-1}(x)=\frac{x}{3}-5\)
  2. \(\displaystyle f^{-1}(x)=\frac{x-5}{3}\)
  3. \(\displaystyle f^{-1}(x)=5(x+3)\)
  4. \(\displaystyle f^{-1}(x)=3x+5\)
(d)
Check your answer to the previous question by finding \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\text{.}\)

Observation 2.6.13.

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

Activity 2.6.14.

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*}
(a)
Interchange the variables \(x\) and \(y\text{.}\)
  1. \(\displaystyle y=\frac{x}{x+1}\)
  2. \(\displaystyle x=\frac{y}{x+1}\)
  3. \(\displaystyle x=\frac{y}{y+1}\)
  4. \(\displaystyle x=\frac{x}{y+1}\)
(b)
Eliminate the denominator.
  1. \(\displaystyle y(x+1)=x\)
  2. \(\displaystyle x(x+1)=y\)
  3. \(\displaystyle x(y+1)=y\)
  4. \(\displaystyle x(y+1)=x\)
(c)
Distribute and gather the \(y\) terms together.
  1. \(\displaystyle yx+y=x\)
  2. \(\displaystyle x^{2}+x=y\)
  3. \(\displaystyle xy-y=-x\)
  4. \(\displaystyle xy=0\)
(d)
Write the inverse function, by factoring and solving for \(y\text{.}\)
  1. \(\displaystyle h^{-1}(x)= \frac{x}{x-1}\)
  2. \(\displaystyle h^{-1}(x)=\frac{x}{1-x}\)
  3. \(\displaystyle h^{-1}(x)= \frac{-x}{1-x}\)
  4. \(\displaystyle h^{-1}(x)= \frac{x+1}{x}\)

Activity 2.6.15.

Find the inverse of each function, using either method. Check your answer using function composition.
(a)
\(g(x)=\dfrac{4x-1}{7} \)
  1. \(\displaystyle g^{-1}(x)=\frac{7x+1}{4} \)
  2. \(\displaystyle g^{-1}(x)= \frac{7x}{4}+1\)
  3. \(\displaystyle g^{-1}(x)= \frac{4x+1}{7}\)
  4. \(\displaystyle g^{-1}(x)=\frac{7}{4x-1} \)
(b)
\(f(x)=3-\sqrt{x+5} \)
  1. \(\displaystyle f^{-1}(x)=3+\sqrt{x-5} \)
  2. \(\displaystyle f^{-1}(x)=(x-3)^{2}+5 \)
  3. \(\displaystyle f^{-1}(x)= \frac{1}{3-\sqrt{x+5}}\)
  4. \(\displaystyle f^{-1}(x)=(3-x)^{2}-5 \)

Subsection 2.6.2 Videos

It would be great to include videos down here, like in the Calculus book!
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