Calculating the inverse of a quadratic function is simple: it is sufficient to make the equation explicit with respect to x and replace y with x in the resulting expression. Finding the inverse of a quadratic function is very misleading, especially since Quadratic functions are not one-to-one functions, except for an appropriate bounded domain.
Steps
Step 1. Explicit with respect to y or f (x) if not already so
During your algebraic manipulations do not modify the function in any way and perform the same operations on both sides of the equation.
Step 2. Arrange the function so that it is of the form y = a (x-h)2+ k.
This is not only critical for finding the inverse of the function, but also for determining whether the function actually has an inverse. You can do this using two methods:
- Completing the square
- "Collect the common factor a" from all terms of the equation (the coefficient of x2). Do this by writing the value of a, opening a parenthesis, and writing the entire equation, then dividing each term by the value of a, as shown in the diagram on the right. Leave the left side of the equation unchanged, as we have not made any actual changes to the right side value.
- Complete the square. The coefficient of x is (b / a). Divide it in half to get (b / 2a), and square it, to get (b / 2a)2. Add it and subtract it from the equation. This will have no modifying effect on the equation. If you look closely, you will see that the first three terms inside the parenthesis are in the form a2+ 2ab + b2, where a is x, so what (b / 2a). Obviously these terms will be numerical and not algebraic for a real equation. This is a completed square.
- Since the first three terms now make up a perfect square, you can write them in the form (a-b)2 o (a + b)2. The sign between the two terms will be the same sign as the coefficient of x in the equation.
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Take the term that is outside the perfect square, from the square brackets. This leads to the equation having the form y = a (x-h)2+ k, as desired.
- Comparing the coefficients
- Create an identity in x. On the left, enter the function as expressed in the form of the x, and on the right enter the function in the desired form, in this case a (x-h)2+ k. This will allow you to find the values of a, h, and k that fit all values of x.
- Open and develop the parenthesis of the right side of the identity. We shouldn't touch the left side of the equation, and we could omit it from our work. Note that all the work that is done on the right hand side is algebraic as shown and not numeric.
- Identify the coefficients of each power of x. Then group them and place them in brackets, as shown on the right.
- Compare the coefficients for each power of x. The coefficient of x2 of the right side must be the same as the one on the left side. This gives us the value of a. The coefficient of x of the right side must be equal to that of the left side. This leads to the formation of an equation in a and in h, which can be solved by replacing the value of a, which has already been found. The coefficient of x0, or 1, of the left side must be the same as that of the right side. By comparing them, we obtain an equation that will help us find the value of k.
- Using the values of a, h, and k found above, we can write the equation in the desired form.
Step 3. Make sure that the value of h is either within the boundaries of the domain, or outside
The value of h gives us the x coordinate of the stationary point of the function. A stationary point within the domain would mean that the function is not bijective, so it does not have an inverse. Note that the equation is a (x-h)2+ k. So if there were (x + 3) inside the parenthesis, the value of h would be -3.
Step 4. Explicit the formula with respect (x-h)2.
Do this by subtracting the value of k from both sides of the equation, and then dividing both sides by a. At this point I would have the numeric values of a, h and k, so use those and not the symbols.
Step 5. Extract the square root of both sides of the equation
This will remove the quadratic power from (x - h). Don't forget to insert the "+/-" sign on the other side of the equation.
Step 6. Decide between the + and - signs, since you can't keep both (keeping both would have a one-to-many "function", which would make it invalid)
To do this, look at the domain. If the domain is to the left of the stationary point eg. x a certain value, use the + sign. Then, make the formula explicit with respect to x.
Step 7. Replace y with x, and x with f-1(x), and congratulate yourself on having successfully found the inverse of a quadratic function.
Advice
- Check your inverse by calculating the value of f (x) for a certain value of x, and then replace that value of f (x) in the inverse to see if the original value of x returns. For example, if the function of 3 [f (3)] is 4, then substituting 4 in the inverse you should get 3.
- If it's not too problematic, you can also check the inverse by analyzing its graph. It should have the same appearance as the original function reflected with respect to the y = x axis.
