Question

If `f ( x ) = 2x - 9 and g ( x ) = ( x + 1) ^ { 2},` what is the composite function `f(g(x))`?

Answer 1

Substitute the functions into one another and get `f(g(x)) = 2(x + 1)^2 - 9 = 2x^2 + 4x - 7`.

Explanation:

We have `f(x) = 2x - 9` and `g(x) = (x+1)^2` and are asked to find `f(g(x)`. Start by putting `g(x)` into that:

`f(g(x)) = f((x+1)^2)`

Now, if `f(x) = 2x - 9`, then anything can fit inside `x` and be put into the equation. Let's set variable `k = (x+1)^2` and fit it inside `f(x)`:

`f((x+1)^2)= f(k) = 2k - 9`

Now we just replace `k` with `(x+1)^2`:

`f(k) = 2k - 9 = 2(x+1)^2 - 9`

Since essentially `k = (x+1)^2 = g(x)`, we now have `f(k) = f(g(x)) = 2(x+1)^2 - 9`.

If asked to expand, do so:

`f(g(x)) = 2(x+1)^2 - 9 = 2(x + 1)(x + 1) - 9`

`= 2(x^2 + x + x + 1) - 9 = 2(x^2 + 2x + 1) - 9`

`= 2x^2 + 4x + 2 - 9 = 2x^2 + 4x - 7`

So we have:

`f(g(x)) = 2(x + 1)^2 - 9 = 2x^2 + 4x - 7`