### Learning Outcomes

- Restrict the domain of a quadratic function, then find its inverse.
- Find the inverse of a radical function.
- Determine the domain of a radical function composed with other functions.
- Find the inverse of a rational function.

So far we have been able to find the inverse functions of **cubic functions** without having to restrict their domains. However, as we know, not all cubic polynomials are one-to-one. Some functions that are not one-to-one may have their domain restricted so that they are one-to-one, but only over that domain. The function over the restricted domain would then have an **inverse function**. Since quadratic functions are not one-to-one, we must restrict their domain in order to find their inverses.

### A General Note: Restricting the Domain

If a function is not one-to-one, it cannot have an inverse function. If we restrict the domain of the function so that it becomes one-to-one, thus creating a new function, this new function will have an inverse function.

### How To: Given a polynomial function, restrict the domain of a function that is not one-to-one and then find the inverse.

- Restrict the domain by determining a domain on which the original function is one-to-one.
- Replace [latex]f(x)[/latex] with [latex]y[/latex].
- Interchange [latex]x[/latex] and [latex]y[/latex].
- Solve for [latex]y[/latex], and rename the function or pair of function [latex]{f}^{-1}\left(x\right)[/latex].
- Revise the formula for [latex]{f}^{-1}\left(x\right)[/latex] by ensuring that the outputs of the inverse function correspond to the restricted domain of the original function.

### recall transforming a quadratic equation

In the examples below, it will be helpful to recall how to transform the graph of a quadratic function. Recall that a quadratic function in the form [latex]f(x)=ax^2+bx+c[/latex] can be written in vertex form, [latex]f(x)=a(x-h)^2+k[/latex], where the vertex [latex]\left(h, k\right)[/latex] can be found by evaluating [latex]\left( - \dfrac{b}{2a}, f\left(- \dfrac{b}{2a}\right)\right)[/latex].

The vertex form of a quadratic also represents the toolkit function [latex]f(x)=x^2[/latex] that has undergone a vertical stretch or compression by a factor of [latex]a[/latex], a horizontal shift [latex]h[/latex] units, and a vertical shift [latex]k[/latex] units.

Ex. The function [latex]f(x) = 3(x-2)^2 -5[/latex] has been stretched vertically by a factor of [latex]3[/latex], shifted [latex]2[/latex] units to the right, and shifted [latex]5[/latex] units down.

### Example: Restricting the Domain to Find the Inverse of a Polynomial Function

Find the inverse function of [latex]f[/latex]:

- [latex]f\left(x\right)={\left(x - 4\right)}^{2}, x\ge 4[/latex]
- [latex]f\left(x\right)={\left(x - 4\right)}^{2}, x\le 4[/latex]

### Example: Finding the Inverse of a Quadratic Function When the Restriction Is Not Specified

Restrict the domain and then find the inverse of

[latex]f\left(x\right)={\left(x - 2\right)}^{2}-3[/latex].

### Try It

Find the inverse of the function [latex]f\left(x\right)={x}^{2}+1[/latex], on the domain [latex]x\ge 0[/latex].

https://ohm.lumenlearning.com/multiembedq.php?id=129081&theme=oea&iframe_resize_id=mom1

Watch the following video to see more examples of how to restrict the domain of a quadratic function to find it’s inverse.

Notice that the functions from previous examples were all polynomials, and their inverses were radical functions. If we want to find the **inverse of a radical function**, we will need to restrict the domain of the answer because the range of the original function is limited.

### How To: Given a radical function, find the inverse.

- Determine the range of the original function.
- Replace [latex]f(x)[/latex] with [latex]y[/latex], then solve for [latex]x[/latex].
- If necessary, restrict the domain of the inverse function to the range of the original function.

### Example: Finding the Inverse of a Radical Function

Determine the range of the function [latex]f\left(x\right)=\sqrt{x - 4}[/latex]. Then, find its inverse and restrict the domain of the inverse as necessary.

### Try It

Determine the range of the function and then find its inverse and appropriate domain. [latex]f\left(x\right)=\sqrt{2x+3}[/latex].

https://ohm.lumenlearning.com/multiembedq.php?id=3333&theme=oea&iframe_resize_id=mom11

## Solving Applications of Radical Functions

Radical functions are common in physical models, as we saw in the section opener. We now have enough tools to be able to solve the problem posed at the start of the section.

### Example: Solving an Application with a Cubic Function

A mound of gravel is in the shape of a cone with the height equal to twice the radius. The volume of the cone in terms of the radius is given by

[latex]V=\frac{2}{3}\pi {r}^{3}[/latex]

Find the inverse of the function [latex]V=\frac{2}{3}\pi {r}^{3}[/latex] that determines the volume [latex]V[/latex] of a cone and is a function of the radius [latex]r[/latex]. Then use the inverse function to calculate the radius of such a mound of gravel measuring 100 cubic feet. Use [latex]\pi =3.14[/latex].

## Determining the Domain of a Radical Function Composed with Other Functions

When radical functions are composed with other functions, determining domain can become more complicated.

### Example: Finding the Domain of a Radical Function Composed with a Rational Function

Find the domain of the function [latex]f\left(x\right)=\sqrt{\dfrac{\left(x+2\right)\left(x - 3\right)}{\left(x - 1\right)}}[/latex].

### Try it

## Finding Inverses of Rational Functions

As with finding inverses of quadratic functions, it is sometimes desirable to find the **inverse of a rational function**, particularly of rational functions that are the ratio of linear functions, such as in concentration applications.

### Example: Finding the Inverse of a Rational Function

The function [latex]C=\dfrac{20+0.4n}{100+n}[/latex] represents the concentration [latex]C[/latex] of an acid solution after [latex]n[/latex] mL of 40% solution has been added to 100 mL of a 20% solution. First, find the inverse of the function; that is, find an expression for [latex]n[/latex] in terms of [latex]C[/latex]. Then use your result to determine how much of the 40% solution should be added so that the final mixture is a 35% solution.

### Try It

Find the inverse of the function [latex]f\left(x\right)=\dfrac{x+3}{x - 2}[/latex]. Verify that you are correct by graphing both functions and the function [latex]f(x) = x[/latex].

https://ohm.lumenlearning.com/multiembedq.php?id=29616&theme=oea&iframe_resize_id=mom5

Watch this video to see another worked example of how to find the inverse of a rational function.