Question

Let `f(x)=2x^2+5x-12` and `g(x)= x+ 4`. Perform the function operation and then find the domain of the function (f* g)?

im confused

Answer 1

`(f@g)(x) = 2x^2+21x+40`

with domain `(-oo, +oo)`

Explanation:

I would guess that you actually mean `f@g` rather than `f*g`

`f@g` denotes the composition of the functions `g` and `f`, defined by:

`(f@g)(x) = f(g(x))`

So it is the function that applies `g` to `x` and then applies `f` to the result.

In our example we have:

`(f@g)(x) = f(g(x))`

`color(white)((f@g)(x)) = f(x+4)`

`color(white)((f@g)(x)) = 2(x+4)^2+5(x+4)-12`

`color(white)((f@g)(x)) = 2(x^2+8x+16)+5(x+4)-12`

`color(white)((f@g)(x)) = 2x^2+16x+32+5x+20-12`

`color(white)((f@g)(x)) = 2x^2+21x+40`

All of `f(x)`, `g(x)` and `(f@g)(x)` are polynomials, so have domain the whole of the real numbers `RR`. In interval notation `(-oo, +oo)`.