What we want below is to uncover the inverse function– which implies that theinverse MUST be a duty itself. Otherwise, we obtained an inverse that is no a function.

You are watching: How to find inverse of a quadratic function

Not allfunctions are normally “lucky” to have actually inversefunctions. This happens in the situation of quadratics because they all fail the Horizontal line Test. However, if ns restrict their domain to whereby the x values develop a graph that would pass the horizontal line test, then i will have an station function.

But first, let’s talk around the test which assures that the inverse is a function.

Horizontal heat Test

Given a function f(x), it has an station denoted through the symbol colorredf^ - 1left( x ight), if no horizontal line intersects that is graph more than one time.

Example the a graph with one inverse
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Key measures in recognize the Inverse function of a Quadratic Function

Replace f(x) through y.Switch the functions of colorredx and colorbluey. In various other words, interchange x and y in the equation.Solve for y in regards to x.Replace y through f^ - 1left( x ight) to get the station functionSometimes, it is beneficial to use the domain and variety of the original duty to identify the exactly inverse functionout of 2 possibilities. This happens when you obtain a “plus or minus” case in the end.
Examples of how to find the Inverse function of a Quadratic Function

Example 1: find the inverse duty offleft( x ight) = x^2 + 2, if the exists. State the domain and also range.

The first thing ns realize is the this quadratic role doesn’t have actually a border on the domain. Ns am certain that as soon as I graph this, i can attract a horizontal heat that will intersect it more than once. Thereforethe station is not a function. I willnot also bother applying the key steps over to find its inverse.

The diagram mirrors that it falls short the Horizontal line Test, for this reason the station is not a function. Ns will prevent here.


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Example 2: discover the inverse role offleft( x ight) = x^2 + 2,,,x ge 0, if the exists. State its domain and range.

This exact same quadratic function, as viewed in instance 1, has a restriction on that domainwhich is x ge 0. After plot the role in xy-axis, I have the right to see the the graph is a parabola reduced in fifty percent for all x values equal to or greater than zero. This have to pass the Horizontal line Test which tells me the I can actually uncover its inverse role by following the suggested steps.

In that is graph below, I plainly defined the domain and selection because ns will require this information to assist me recognize the correct inverse role in the end.

Remember that the domain and selection of the inverse duty come native the range, and domain that the initial function, respectively. It’s called the swapping that domain and also range.


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Even without solving for the inverse function just yet, i can quickly identify the domain and range using the details from the graph that the initial function: domain is x ≥ 2 and also range is y ≥ 0.

Do friend see exactly how I interchange the domain and selection of the original role to get the domain and variety of the inverse?

Now, let’s walk ahead and algebraically deal with for that is inverse.


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Example 3: discover the inverse function of fleft( x ight) = - x^2 - 1,,,x le 0, if the exists. State its domain and range.

This problem is very similar to instance 2. The range starts at colorredy=-1, and it deserve to go down as lowas possible.


Now, these room the actions on just how to deal with for the inverse.

Applying square root operation results in getting two equationsbecause of the confident and an adverse cases. To pick the exactly inverse duty out of the two, I indicate that you discover the domain and range of each feasible answer. Now, the correct inverse duty should have a domain comes from the selection of the original function; and also a variety coming indigenous the domain of the exact same function.


I would graph this function very first and plainly identify the domain and range. Notification that the border in the domain cut the parabola right into two equal halves.I will resolve the left half that this parabola. Clearly, this has actually an inverse function because it passes the Horizontal heat Test.


Proceed v the actions in addressing for the inverse function. In fact, there are two ways exactly how to occupational this out.

Solve this by the Quadratic Formula as presented below.

This is expected because we are solving for a function, not precise values.

The vital step below is to pick the proper inverse duty in the endbecause us will have actually the add to (+) and minus (−) cases. We deserve to do the by recognize the domain and variety of each and also compare the to the domain and variety of the initial function. Rememberthat we swapthe domain and selection of the original role to obtain the domain and selection of its inverse.


If you observe, the graphs that the duty and that inverse space actually symmetrical along the line y = x (see dashed line). They are like mirror photos of eachother.

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I hope that you acquire some level of appreciation on exactly how to discover the inverse of aquadratic function. Although it can be a little bit tedious, as you can see, overall it is no that bad. I recommend that you inspect out the connected lessons on exactly how to find inverses of other kinds the functions.

Practice through Worksheets


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MATH SUBJECTSIntroductory AlgebraIntermediate AlgebraAdvanced AlgebraAlgebra word ProblemsGeometryIntro to Number TheoryBasic mathematics Proofs
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