Question

How do you find the domain of f+g given f(x)= x^2 - 2x and g(x)= x-5?

Answer 1

the domain of any quadratic equation is all real numbers. `RR`

Explanation:

Quadratic functions are parabolas that stretch infinitely in both directions of the axes.

So for any quadratic equation the Domain is always going to be negative infinity, positive infinity.

interval notation (`- oo, oo`)

set builder = `{x| x in RR`}

Answer 2

Since both `f(x)` and `g(x)` are defined for all Real values:
Domain of `f(g(x))` is `RR`

Explanation:

...there is no real need to evaluate `f(g(x))` in simplified form in terms of just `x`; but here it is anyway:

`f(x)=x^2-2x`

`f(g(x))=g(x)^2-2g(x)`

and since `g(x)=x-5`
`f(g(x))=(x-5)^2-2(x-5)`

`color(white)("XXX")=(x^2-10x+25)-(2x-10)`

`color(white)("XXX")=x^2-12x+35`

(which is defined for all `RR`