Question

How do you find the compositions given `f(x)=3x²` and `g(x)=4-5x`?

Answer 1

Many different compositions are possible. I'll give you one.

Explanation:

The example I'll show you is `f(g(x))`. This means plugging in function g for x into f.

`f(g(x)) = 3(4 - 5x)^2`

`f(g(x)) = 3(16 - 40x + 25x^2)`

`f(g(x)) = 48 - 120x + 75x^2`

`f(g(x)) = 75x^2 - 120x + 48`

Note that function compositions will often be noted as `(f @ g)(x)`, which means the same thing as `f(g(x))` (it must be evaluated from the inside outside; if you have `f(g(h(x)))` for example, you must plug h into g and then plug the answer of that into f.)

When the x in parentheses is replaced by a number, you must plug this in for x.

Ex: if `f(x) = 3x + 4` and `g(x) = 3^x`, find the value of `f(g(2))`.

`f(g(2) = 3(3^2) + 4`

`f(g(2)) = 27 + 4`

`f(g(2)) = 31`

Practice exercises:

1. Find the following compositions if `f(x) = 2x + 5, g(x) = 2x^2 - 4x + 7 and h(x) = sqrt(5x - 1)`

a) `h(f(x))`

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

c). `g(f(h(13)))`

Good luck!

Answer 2

f(g(x)) `= 75x^2 -120x + 48`
g(f(x)) `= 4 - 15x^2`

Explanation:

(1) find f(g(x)) = f(4 - 5x)

To find the value , substitute x = 4 - 5x in for x in f(x)

hence : f(4-5x) = 3(4-5x)^2 = `3(16-40x+25x^2)`

`= 48-120x +75x^2 = 75x^2 -120x + 48`

(2) find g(f(x)) `= g(3x^2)`

To find the value , substitute x = `3x^2" in for x in g(x) "`

hence : `g(3x^2) = 4 - 5(3x^2) = 4 - 15x^2`