Question

Given `f(x) = -3x^2 - 8` and `h(x) = 6x + 2` how do you find f(h(x))?

Answer 1

This means to plug function h in the place of x into f.

Explanation:

`f(h(x)) = -3(6x + 2)^2 - 8`

`f(h(x)) = -3(36x^2 + 24x + 4) - 8`

`f(h(x)) = -108x^2 - 72x - 12 - 8`

`f(h(x)) = -108x^2 - 72x - 20`

Note that `f(h(x)) = (f @ g)(x)`, they're just different notation forms.

When a number replaces the x in parentheses you must plug that number in for x in the inner function, find the result of that calculation, and then plug that in for x in the outer function. When there are three or more functions, make sure to work from the inside out, in the functions' respective order.

Practice exercises:

Assuming that `f(x) = -2x + 5, g(x) = 2x^2 + 3x - 2 and h(x) = sqrt(4x - 3)`, find the following compositions.

a) `h(f(g(x)))`

b) `g(h(f(x)))`

c). `f(h(f(h(x))))`

d). `g(f(h(7)))`