107,075 views
23 votes
23 votes
Given f(x) = 1/2x-4 and g(x)=2x+8, f(x) and g(x) are inverses of each other?Find (f o g ) (x) =and (g o f ) (x) =

Given f(x) = 1/2x-4 and g(x)=2x+8, f(x) and g(x) are inverses of each other?Find (f-example-1
User John Retallack
by
3.2k points

1 Answer

12 votes
12 votes

Answer

- f(x) and g(x) are inverses of each other.

- (f o g) (x) = x

- (g o f) (x) = x

Step-by-step explanation

We are given two functions,

f(x) = ½x - 4

g(x) = 2x + 8

We are firstly asked if the two functions are inverse functions of each other.

Then, we are asked to find (f o g) (x)

And (g o f) (x)

First of, the inverse of a function is a function that reverses the actions of the function. That is, for the inverse function, if we are given f(x), we would be able to obtain x.

The step to finding a function's inverse is to write y instead of f(x) and then make x the subject of formula. We then rewrite the solution from all of this by replacing the y by x and the x, which is a subject of formula now, represents the inverse function, f⁻¹(x).

So,

f(x) = ½x - 4

y = ½x - 4

Multiply through by 2

2y = x - 8

x = 2y + 8

f⁻¹(x) = 2x + 8

This inverse of f(x) is equal to g(x).

f⁻¹(x) = g(x)

Hence, f(x) and g(x) are inverses of each other.

We can then solve for (f o g) (x)

(f o g) (x) means that we need to write f(x), but intead of x, we replace x with g(x).

(f o g) (x) = f[g(x)]

f(x) = ½x - 4

f[g(x)] = ½[g(x)] - 4

= ½(2x + 8) - 4

= x + 4 - 4

= x

(g o f) (x) means that we need to write g(x), but intead of x, we replace x with f(x).

g(x) = 2x + 8

g[f(x)] = 2[f(x)] + 8

= 2(½x - 4) + 8

= x - 8 + 8

= x

Hope this Helps!!!

User Mounir
by
3.4k points