### Learning Outcomes

- Evaluate composite functions
- Use compositions of functions to algebraically verify inverses
- Identify the domain and range of inverse functions
- Interpret the meaning of [latex]f^{-1}(x)[/latex]

Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.

Using descriptive variables, we can notate these two functions. The function [latex]C\left(T\right)[/latex] gives the cost [latex]C[/latex] of heating a house for a given average daily temperature in [latex]T[/latex] degrees Celsius. The function [latex]T\left(d\right)[/latex] gives the average daily temperature on day [latex]d[/latex] of the year. For any given day, [latex]\text{Cost}=C\left(T\left(d\right)\right)[/latex] means that the cost depends on the temperature which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature [latex]T\left(d\right)[/latex]. For example, we could evaluate [latex]T\left(5\right)[/latex] to determine the average daily temperature on the [latex]5[/latex]th day of the year. Then, we could evaluate the **cost function** at that temperature. We would write [latex]C\left(T\left(5\right)\right)[/latex]. By combining these two relationships into one function, we have performed function composition.

We read the left-hand side as [latex]``f[/latex] composed with [latex]g[/latex] at [latex]x,''[/latex] and the right-hand side as [latex]``f[/latex] of [latex]g[/latex] of [latex]x.''[/latex] The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol [latex]\circ [/latex] is called the composition operator.

It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first and then working to the outside.

### Example

Using the functions provided, find [latex]f\left(g\left(x\right)\right)[/latex] and [latex]g\left(f\left(x\right)\right)[/latex].

[latex]f\left(x\right)=2x+1[/latex]

[latex]g\left(x\right)=3-x[/latex]

In the following video, you will see another example of how to find the composition of two functions.

## Inverse Functions

An **inverse function** is a function where the input of the original function becomes the output of the inverse function. This naturally leads to the output of the original function becoming the input of the inverse function. The reason we want to introduce inverse functions is because exponential and logarithmic functions are inverses of each other, and understanding this quality helps to make understanding logarithmic functions easier. And the reason we introduced composite functions is because you can verify, algebraically, whether two functions are inverses of each other by using a composition.

Given a function [latex]f\left(x\right)[/latex], we represent its inverse as [latex]{f}^{-1}\left(x\right)[/latex], read as “[latex]f[/latex] inverse of [latex]x.[/latex]” The raised [latex]-1[/latex] is part of the notation. It is not an exponent; it does not imply a power of [latex]-1[/latex] . In other words, [latex]{f}^{-1}\left(x\right)[/latex] does *not* mean [latex]\frac{1}{f\left(x\right)}[/latex] because [latex]\frac{1}{f\left(x\right)}[/latex] is the reciprocal of [latex]f[/latex] and not the inverse.

[Note: If you did want to write [latex]\frac{1}{f\left(x\right)}[/latex] using an exponent, you would write it as [latex]\frac{1}{f\left(x\right)}=(f(x))^{-1}[/latex].]

Just as zero does not have a **reciprocal**, some functions do not have inverses.

### Inverse Function

For any **one-to-one function** [latex]f\left(x\right)=y[/latex], a function [latex]{f}^{-1}\left(x\right)[/latex] is an **inverse function** of [latex]f[/latex] if [latex]{f}^{-1}\left(y\right)=x[/latex].

The notation [latex]{f}^{-1}[/latex] is read “[latex]f[/latex] inverse.” Like any other function, we can use any variable name as the input for [latex]{f}^{-1}[/latex], so we will often write [latex]{f}^{-1}\left(x\right)[/latex], which we read as “[latex]f[/latex] inverse of [latex]x.''[/latex]

Keep in mind that

and not all functions have inverses.

In our first example, we will identify an inverse function from ordered pairs.

### Example

If for a particular one-to-one function [latex]f\left(2\right)=4[/latex] and [latex]f\left(5\right)=12[/latex], what are the corresponding input and output values for the inverse function?

### Analysis of the Solution

In the following video, we show an example of finding corresponding input and output values given two ordered pairs from functions that are inverses.

### How To: Given two functions [latex]f\left(x\right)[/latex] and [latex]g\left(x\right)[/latex], test whether the functions are inverses of each other

- Substitute [latex]g(x)[/latex] into [latex]f(x)[/latex]. The result must be x. [latex]f\left(g(x)\right)=x[/latex]
- Substitute [latex]f(x)[/latex] into [latex]g(x)[/latex]. The result must be x. [latex]g\left(f(x)\right)=x[/latex]

If [latex]f(x)[/latex] and [latex]g(x)[/latex] are inverses, then [latex]f(x)=g^{-1}(x)[/latex] and [latex]g(x)=f^{-1}(x)[/latex]

In our next example, we will test inverse relationships algebraically.

### Example

If [latex]f\left(x\right)=x^2-3[/latex], for [latex]x\ge0[/latex] and [latex]g\left(x\right)=\sqrt{x+3}[/latex], is [latex]g[/latex] the inverse of [latex]f[/latex]? In other words, does [latex]g={f}^{-1}?[/latex]

In the following video, we use algebra to determine if two functions are inverses.

We will show one more example of how to verify whether you have an inverse algebraically.

### Example

If [latex]f\left(x\right)=\frac{1}{x+2}[/latex] and [latex]g\left(x\right)=\frac{1}{x}-2[/latex], is [latex]g[/latex] the inverse of [latex]f[/latex]? In other words, does [latex]g={f}^{-1}?[/latex]

We will show one more example of how to use algebra to determine whether two functions are inverses of each other.

## Domain and Range of a Function and Its Inverse

The outputs of the function [latex]f[/latex] are the inputs to [latex]{f}^{-1}[/latex], so the range of [latex]f[/latex] is also the domain of [latex]{f}^{-1}[/latex]. Likewise, because the inputs to [latex]f[/latex] are the outputs of [latex]{f}^{-1}[/latex], the domain of [latex]f[/latex] is the range of [latex]{f}^{-1}[/latex]. We can visualize the situation.

In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function [latex]f\left(x\right)={x}^{2}[/latex] with its domain limited to [latex]\left[0,\infty \right)[/latex], which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).

### Domain and Range of Inverse Functions

The range of a function [latex]f\left(x\right)[/latex] is the domain of the inverse function [latex]{f}^{-1}\left(x\right)[/latex].

The domain of [latex]f\left(x\right)[/latex] is the range of [latex]{f}^{-1}\left(x\right)[/latex].

In our last example, we will define the domain and range of a function’s inverse using a table of values, and evaluate the inverse at a specific value.

### Example

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.

- Define the domain and range of the function and its inverse.
- Find and interpret [latex]{f}^{-1}\left(70\right)[/latex].

[latex]t\text{ (minutes)}[/latex] |
[latex]30[/latex] | [latex]50[/latex] | [latex]70[/latex] | [latex]90[/latex] |

[latex]f\left(t\right)\text{ (miles)}[/latex] |
[latex]20[/latex] | [latex]40[/latex] | [latex]60[/latex] | [latex]70[/latex] |

## Summary

The inverse of a function can be defined for one-to-one functions. If a function is not one-to-one, it can be possible to restrict its domain to make it so. The domain of a function will become the range of its inverse. The range of a function will become the domain of its inverse. Inverses can be verified using tabular data as well as algebraically.

Notice that if we show the coordinate pairs from the previous example in table form, the input and output are clearly reversed.