Rational and Radical Functions: Background You’ll Need 3
Discover or calculate the inverse of a function
Finding and Evaluating Inverse Functions
Recall that two functions [latex]f[/latex] and [latex]g[/latex] are inverse functions if for every coordinate pair in [latex]f[/latex], [latex](a, b)[/latex], there exists a corresponding coordinate pair in the inverse function, [latex]g[/latex], [latex](b, a)[/latex]. In other words, the coordinate pairs of the inverse functions have the input and output interchanged.
Only one-to-one functions have inverses that are also functions. Recall that a one-to-one function has a unique output value for each input value and passes the horizontal line test.
For a function that is not one-to-one to have an inverse function, the function must be restricted on its domain to create a new function that is one-to-one and thus would have an inverse function.
verifying two functions are inverses of one another
Two functions, [latex]f[/latex] and [latex]g[/latex], are inverses of one another if for all [latex]x[/latex] in the domain of [latex]f[/latex]and [latex]g[/latex].
Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.
Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.
A function [latex]f\left(t\right)[/latex] is given below, showing distance in miles that a car has traveled in [latex]t[/latex] minutes. Find and interpret [latex]{f}^{-1}\left(70\right)[/latex].
[latex]t[/latex] (minutes)
[latex]f(t)[/latex] (miles)
[latex]30[/latex]
[latex]20[/latex]
[latex]50[/latex]
[latex]40[/latex]
[latex]70[/latex]
[latex]60[/latex]
[latex]90[/latex]
[latex]70[/latex]
The inverse function takes an output of [latex]f[/latex] and returns an input for [latex]f[/latex]. So in the expression [latex]{f}^{-1}\left(70\right)[/latex], [latex]70[/latex] is an output value of the original function, representing [latex]70[/latex] miles. The inverse will return the corresponding input of the original function [latex]f[/latex], [latex]90[/latex] minutes, so [latex]{f}^{-1}\left(70\right)=90[/latex]. The interpretation of this is that, to drive [latex]70[/latex] miles, it took [latex]90[/latex] minutes.
Alternatively, recall that the definition of the inverse was that if [latex]f\left(a\right)=b[/latex], then [latex]{f}^{-1}\left(b\right)=a[/latex]. By this definition, if we are given [latex]{f}^{-1}\left(70\right)=a[/latex], then we are looking for a value [latex]a[/latex] so that [latex]f\left(a\right)=70[/latex]. In this case, we are looking for a [latex]t[/latex] so that [latex]f\left(t\right)=70[/latex], which is when [latex]t=90[/latex].
Evaluating the Inverse of a Function, Given a Graph of the Original Function
The domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.
How To: Given the graph of a function, evaluate its inverse at specific points.
Find the desired input of the inverse function on the [latex]y[/latex]-axis of the given graph.
Read the inverse function’s output from the [latex]x[/latex]-axis of the given graph.
A function [latex]g\left(x\right)[/latex] is given below. Find [latex]g\left(3\right)[/latex] and [latex]{g}^{-1}\left(3\right)[/latex].
To evaluate [latex]g\left(3\right)[/latex], we find [latex]3[/latex] on the [latex]x[/latex]-axis and find the corresponding output value on the [latex]y[/latex]-axis. The point [latex]\left(3,1\right)[/latex] tells us that [latex]g\left(3\right)=1[/latex].
[latex]\\[/latex]
To evaluate [latex]{g}^{-1}\left(3\right)[/latex], recall that by definition [latex]{g}^{-1}\left(3\right)[/latex] means the value of [latex]x[/latex] for which [latex]g\left(x\right)=3[/latex]. By looking for the output value 3 on the vertical axis, we find the point [latex]\left(5,3\right)[/latex] on the graph, which means [latex]g\left(5\right)=3[/latex], so by definition, [latex]{g}^{-1}\left(3\right)=5[/latex].
Finding Inverses of Functions Represented by Formulas
Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula—for example, [latex]y[/latex] as a function of [latex]x-[/latex] we can often find the inverse function by solving to obtain [latex]x[/latex] as a function of [latex]y[/latex].
How To: Given a function represented by a formula, find the inverse.
Verify that [latex]f[/latex] is a one-to-one function.
Replace [latex]f\left(x\right)[/latex] with [latex]y[/latex].
Interchange [latex]x[/latex] and [latex]y[/latex].
Solve for [latex]y[/latex], and rename the function [latex]{f}^{-1}\left(x\right)[/latex].
Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.
[latex]C=\frac{5}{9}\left(F - 32\right)[/latex]
[latex]{ C }=\frac{5}{9}\left(F - 32\right)[/latex]
[latex]C\cdot \frac{9}{5}=F - 32[/latex]
[latex]F=\frac{9}{5}C+32[/latex]
By solving in general, we have uncovered the inverse function. If
In this case, we introduced a function [latex]h[/latex] to represent the conversion because the input and output variables are descriptive, and writing [latex]{C}^{-1}[/latex] could get confusing.
Find the inverse of the function [latex]f\left(x\right)=\dfrac{2}{x - 3}+4[/latex].
[latex]\begin{align}&y=\frac{2}{x - 3}+4 && \text{Change }f(x)\text{ to }y. \\[1.5mm]&x=\frac{2}{y - 3}+4 && \text{Switch }x\text{ and }y. \\[1.5mm] &y - 4=\frac{2}{x - 3} && \text{Subtract 4 from both sides}. \\[1.5mm] &y - 3=\frac{2}{x - 4} && \text{Multiply both sides by }y - 3\text{ and divide by }x - 4. \\[1.5mm] &y=\frac{2}{x - 4}+3 && \text{Add 3 to both sides}.\\[-3mm]&\end{align}[/latex]
So [latex]{f}^{-1}\left(x\right)=\dfrac{2}{x - 4}+3[/latex].
Analysis of the Solution
The domain and range of [latex]f[/latex] exclude the values [latex]3[/latex] and [latex]4[/latex], respectively. [latex]f[/latex] and [latex]{f}^{-1}[/latex] are equal at two points but are not the same function, as we can see by creating the table below.
