Q&A for How to Find the Inverse of a Function

The 5's cancel each other out during the process. Here is the extended working out. 3a + 5 = 3b + 5, 3a +5 -5 = 3b, 3a = 3b.

First convert from f(x+3) into the form f(x), call it g(x) = 3(x - 3) + 5 = 3x - 9 + 5 = 3x - 4. Let g(x) = y. Then y = 3x - 4. Switch around: x = 3y - 4. x + 4 = 3y, so y = (x + 4)/3. So the inverse function is f⁻¹(x) = (x + 4)/3

Set this equal to f(x). Then replace this with y. Then switch the two letters around. x = 3y^2 + 5y/2y + 4y^2. Then multiply both sides by the denominator. x(2y + 4y^2) = 3y + 2. 2y + 4y^2 = 3y + 2 / x. Then 2y = 3y + 2 / x - 4y^2. and y = 3y + 2 / x - 4y^2 / 2.

Let g(x) = y. Then switch x and y around. x = 2y^2 - 4. Then x + 4 = 2y^2. x + 4 / 2 = y^2, or x/2 + 2 = y^2. Then y = the square root of x/2 + 2. Then replace the inverse of g(x) with this, and you've got your answer.

If we let f(x) = y, then this becomes y = 3x + 5. Then if we switch x and y around, we get x = 3y + 5. Then x - 5 = 3y, and y = (x - 5) / 3. Finally replace y with your inverse function which is (x - 5) / 3.