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!!!