Determine the inverse function of 𝑓 of 𝑥 equals 𝑥 plus six all squared minus five, where 𝑥 is greater than or equal to, more than or equal to, negative six.
We’re going to look at two different methods to find the inverse function. Our first method involves following these four steps. Step one, replace 𝑓 of 𝑥 with 𝑦. In this case, it gives us 𝑦 equals 𝑥 plus six all squared minus five. Step two, replace every 𝑥 with a 𝑦 and every 𝑦 with an 𝑥. The equation now reads 𝑥 equals 𝑦 plus six all squared minus five.
Step three, make 𝑦 the subject of the formula, or solve for 𝑦 equals. In order to do this step, we’re gonna have to balance the equation. Our first step is to add five to both sides. This leaves us with 𝑥 plus five equals 𝑦 plus six all squared. Square rooting both sides of the equation leaves us with the square root of 𝑥 plus five equals 𝑦 plus six.
And finally, subtracting six from both sides of the equation gives us the square root of 𝑥 plus five take away six equals 𝑦. Step five is just to replace the 𝑦 with 𝑓 minus one of 𝑥, the inverse of 𝑓. Therefore, the inverse of 𝑓, 𝑓 minus one of 𝑥, is equal to the square root of 𝑥 plus five minus six.
An alternate method to find the inverse function of 𝑓 of 𝑥 equals 𝑥 plus six all squared minus five is to use function machines. What three operations are we doing to the 𝑥? Well firstly, we are adding six to it. Secondly, we are squaring that answer. And finally, we are subtracting five.
Let’s now think about what happens when we do the inverse operations. The opposite or inverse of subtracting five is adding five. The inverse operation to squaring a number is square rooting it. And finally the inverse operation of adding six is subtracting six.
If we substitute an 𝑥 to these function machines, we would add five, we would square root the answer, and then we would subtract six. This means that our inverse function, 𝑓 minus one of 𝑥, is the square root of 𝑥 plus five minus six. This is the same answer as using the first method.
In order to check that our answer is correct, we can try substituting in some values. Let’s work out 𝑓 of two. Two add six gives us eight. Eight squared is 64. Subtracting five from this gives us an answer of 59. Substituting 59 into our inverse function, 𝑓 minus one of 𝑥 equals the square root of 𝑥 plus five minus six should hopefully give us an answer of two.
59 plus five is 64. The square root, or root, of 64 is eight. Eight minus six is equal to two. As the input to 𝑓 is equal to the output of the inverse function and vice versa, we know that our functions 𝑓 minus one of 𝑥 equals the square root of 𝑥 plus five minus six is correct.