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Related Lessons: Inverse of Linear Function Inverse of Quadratic Function Inverse of Square Root Function Inverse of Rational Function
  Verifying Two Functions are Inverses Inverse of Logarithmic Function Inverse of Exponential Function Inverse of Absolute Value Function
  Inverse of Constant Function      

 

Finding the Inverse Function of a Quadratic Function | Step by Step

 

What we want here is to find the inverse function - which implies that the inverse MUST be a function itself. Otherwise, we got an inverse that is not a function. Not good!

Not all functions are naturally "lucky" to have inverse functions. This happens in the case of quadratics because they all fail the Horizontal Line Test. However, if I restrict their domain to where the x values produce a graph that would pass the horizontal line test, then I will have an inverse function.

But first, let's talk about the test which guarantees that the inverse is a function. It's called the Horizontal Line Test.

Horizontal Line Test

Given a function f(x), it has an inverse denoted by the symbol f −1(x), if no horizontal line intersects its graph more than one time.

Example of graph with an inverse Example of graph without an inverse

diagram showing an example when a graph has an inverse for passing the horizontal line test.

graph showing where the horizontal line test fails

 

The following are the main strategies to algebraically solve for the inverse function.

Key steps

1) Replace f(x) by y
2) Switch the roles of "x" and "y" , in other words, interchange x and y in the equation.
3) Solve for y in terms of x
4) Replace y by f1(x) to get the inverse function
5) *Sometimes, it is helpful to use the domain and range of the original function to identify the correct inverse function out of two possibilities. This happens when you get a "plus or minus" case in the end.

 


Direction: Find the inverse functions of the following quadratic functions, if they exist. In addition, state their domain and range.

1) example 1: f(x)=x^2+2 See solution
2) example 2: f(x)=x^2+2 for x≥0 See solution
3) example 3: f(x)=-x^2-1, for x≤0 See solution
4) example 4: f(x)=x^2+2x-3, for x≤-1 See solution

 


Example 1: Find the inverse function of f(x)=x^2+2 , if it exists. State its domain and range.

The first thing I realize is that this quadratic function doesn't have restriction on its domain. I am sure that when I graph this, I can draw a horizontal line that will intersect it more than once. Therefore the inverse is not a function. I will not even bother applying the key steps above to find its inverse.

The diagram shows that it fails the Horizontal Line Test, thus the inverse is not a function. I will stop here.

showing the graph of f(x)=x^2+2  where it fails the horizontal line test

 


Example 2: Find the inverse function of  f(x)=x^2+2 for x≥0 , if it exists. State its domain and range.

This same quadratic function, as seen in Example 1, has restriction on its domain which is x > 0. After plotting the function in xy-axis, I can see that the graph is a parabola cut in half for all x values equal to or greater than zero. This should pass the Horizontal Line Test which tells me that I can actually find its inverse function by following the suggested steps.

In its graph below, I clearly defined the domain and range because I will need this information to help me identify the correct inverse function in the end.

Remember that the domain and range of the inverse function come from the range, and domain of the original function, respectively. It's called the swapping of domain and range.

diagram showing the domain and range of f(x)=x^2+2 for x greater than or equal to zero

Even without solving for the inverse function just yet, I can easily identify its domain and range using the information from the graph of the original function: domain is x > 2 and range is y > 0.

Do you see how I interchange the domain and range of the original function to get the domain and range of its inverse?

Now, let's go ahead and algebraically solve for its inverse.

for f(x)=x^2+2, its inverse f^-(x)=√(x-2)

 

Graphing the original function with its inverse in the same coordinate axis...

 

graphs of both f(x) and f^-(x)

 


Example 3: Find the inverse function of f(x)=-x^2-1, for x≤0 , if it exists. State its domain and range.

This problem is very similar to Example 2. The range starts at y = −1 and can down as low as possible.

graph showing the domain and range of f(x)=-x^2-1 for x less then or equal to zero

Now, these are the steps on how to solve for the inverse.

Applying square root operation results in getting two equations because of the positive and negative cases. To pick the correct inverse function out of the two, I suggest that you find the domain and range of each possible answers. Now, correct inverse function should have the a domain coming from the range of the original function; and a range coming from the domain of the same function.

inverse of f(x) is f^-1(x)=-√(-x-1)

 

The following are the graphs of the original function and its inverse on the same coordinate axis.

graphs of f(x) and its inverse f^-1(x) in the same coordinate axes (cartesian plane)

 


Example 4: Find the inverse function of f(x)=x^2+2x-3, for x≤-1 , if it exists. State its domain and range.

I would graph this function first, and clearly identify the domain and range. Notice that the restriction in the domain cuts the parabola into two equal halves. I will deal with the left half of this parabola. Clearly this has an inverse function because it passes the Horizontal Line Test.

graph showing half of f(x)=x^2+2x-3

Proceed with the steps in solving for the inverse function. In fact, there are two ways how to work this out. Click the method you want to see the detailed solution.

Solve this by the Quadratic Formula, the quadratic formula: x=[-b±√(b^2-4ac)]/2a, where the values of "a", "b", and "c" can contain variables. This is expected since we are solving for a function , not exact values.

The key step here is to pick the appropriate inverse function in the end because we will have the plus (+) and minus (− ) cases. We can do that by finding the domain and range of each, and compare that to the domain and range of the original function. Remember that we swap the domain and range of the original function to get the domain and range of its inverse.

 

inverse of f(x) = x^2+2x-3 is f^-1(x)= -1-√(x+4)

 

The method of Completing the Squares allows us to isolate the variable in a quadratic trinomial. As you will see in the steps, the quadratic trinomial is converted into linear binomial raised to the 2nd power. Obviously, we can apply the square root operation to get rid of the power 2, therefore leaving us with an easy equation to solve.

 

solving for the inverse of f(x) = x^2+2x-3 using the Completing the Square Method

 

If you observe, the graphs of the function and its inverse is actually symmetrical along the line y = x (see dashed line). They are like mirror images of each other.

 

graphs of f(x) and f^-1(x) in the same cartesian plane. they have symmetry along the line y=x.

 

I hope that you gain some level of appreciation how to find the inverse of a quadratic function. Although it can be a bit tedious, overall it is not that bad. Right? I recommend that you check out the related lessons how to find inverses of  other kinds of functions. Go top of the page.

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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