Let f(x) = 3x -5 and g(x) = -4x +7. Find (fog)(-4)

Question

I really don't understand inverse functions at all so any explanation would be greatly appreciated

Answer 1

Well these are not inverse functions, they are COMPOSITE functions.

(f o g)(x) = f(g(x)) = 3 ( -4x + 7) - 5 <--- substitutes g INSIDE of f

= -12x + 21 - 5

= -12x + 16

So (f o g) (-4) = (-12)(-4) + 16 = 48 + 16 = 64

Note that g(-4) = (-4)(-4) + 7 = 16 + 7 = 23 and f(23) = 3(23)-5 = 69-5 = 64

Now for the INVERSE function.

Finds the inverse of y = f(x) = 3x - 5

STEP 1: SWAPS X and Y: x = 3y - 5

STEP 2: solves for x: x + 5 = 3y

(x+5)/3 = y

So the inverse is y = (x+5)/3

NOTE that when we compose f and its inverse we get:

3 [( x+5)/3] - 5

= x+5-5 <--- 3 cancels

= x

and

inverse compose f : [ (3x-5)+5] /3 = (3x)/3 = x

We get the identity function y=x

THIS MUST HAPPEN in order for the inverse to be correct