# How to Find the Inverse of a Quadratic Function

Calculating the inverse of a linear function is easy: just make x the subject of the equation, and replace y with x in the resulting expression. Finding the inverse of a quadratic function is considerably trickier, not least because Quadratic functions are not, unless limited by a suitable domain, one-one functions.

## Steps

1. 1
Make y or f(x) the subject of the formula if it isn't already. During your algebraic manipulation, make sure that you do not change the function in any way and perform the same operations to both "sides" of the equation.
2. 2
Rearrange the function so that it is in the form y=a(x-h)2+k. This is not only essential for you to find the inverse of the function, but also for you to determine whether the function even has an inverse. You can do this by two methods:
• By completing the square

1. "Take common" from the whole equation the value of a (the coefficient of x2). Do this by writing the value of a, starting a bracket, and writing the whole equation, then dividing each term by the value of a, as shown in the diagram on the right. Leave the left hand side of the equation untouched, as there has been no net change to the right hand side.
2. Complete the square. The coefficient of x is (b/a). Halve it, to give (b/2a), and square it, to give (b/2a)2. Add and subtract it from the equation. This will have no net effect on the equation. If you look closely, you will see that the first three terms inside the bracket are in the form a2+2ab+b2, where a is x, and b is (b/2a). Of course these two values will be numerical, rather than algebraic for a real equation. This is a completed square.
3. Because the first three terms are now a perfect square, you can write them in the form (a-b)2 or (a+b)2. The sign between the two terms will be the same as the sign of the coefficient of x in the equation.
4. Take the term which is outside the perfect square out of the square bracket. This brings the equation into the form y=a(x-h)2+k, as intended.

• By comparing coefficients

1. Form an identity in x. On the left, put the function as it is expressed in terms of x, and on the right put the function in the form that you want it to be, in this case a(x-h)2+k. This will enable you to find out the values of a, h, and k that are true for all values of x.
2. Open and expand the bracket on the right hand side of the identity. We shall not be touching the left hand side of the equation, and may omit it from our working. Note that all working on the right hand side is algebraic as shown and not numerical.
3. Identify the coefficients of each power of x. Then group them and place them in brackets, as shown on the right.
4. Compare the coefficients of each power of x. The coefficient of x2 on the right hand side must equal that on the left hand side. This gives the value of a. The coefficient of x on the right hand side also must equal that on the left hand side. This leads to the formation of an equation in a and h, which can be solved by substituting the value of a, which has already been found. The coefficient of x0, or 1, on the left hand side must also equal that on the right hand side. Comparing them yields an equation that will help us find the value of k.
5. Using the values of a,h, and k found above, we can write the equation in the desired form.
3. 3
Ensure that the value of h is either on the boundary of the domain, or outside it. The value of h gives the x-coordinate of the turning point of the equation. A turning point within the domain would mean that the function is not one-one, and hence does not have an inverse. Note that the equation is a(x-h)2+k. Thus if there is (x+3) inside the bracket, the value of h is negative 3.
4. 4
Make (x-h)2 the subject of your formula. Do this by subtracting the value of k from both sides of the equation, and then dividing both sides of the equation by a. By now you will have numerical values for a,h, and k, so use those, not the symbols.
5. 5
Square-Root both sides of the equation. This will remove the power of two from (x-h). Do not forget to put the "+/-" sign on the other side of the equation.
6. 6
Decide between the + and the - sign, as you can not have both (having both would make it a one to many "function", which would make it invalid as the same). For this, look at the domain. If the domain lies to the left of the stationary point i.e. x < a certain value, use the - sign. If the domain lies to the right of the stationary point i.e. x > a certain value, use the + sign. Then, make x the subject of the formula.
7. 7
Replace y with x, and x with f-1(x), and congratulate yourself on having successfully found the inverse of a quadratic function.

## Tips

• Check your inverse by calculating the value of f(x) for a certain value of x, and then substituting that value of f(x) in the inverse to see if it returns the original value of x. For example, if the function of 3 [f(3)] is 4, then substituting 4 in the inverse should return 3.
• If it is not too much trouble you can also check the inverse by inspecting its graph. It should look like the original function reflected across the line y=x.

## Article Info

Categories: Algebra