Question

if $f(x) = x^2 - 1$ and $g(x) = 2x - 3$, what is the domain of $(f circ g)(x)$?
$\bigcirc\\ (-infty, infty)$
$\bigcirc\\ -1, infty)$
$\bigcirc\\ -5, infty)$
$\bigcirc\\ (infty, -infty)$

Explanation:

Step1: Find composite function $(f\circ g)(x)$

Substitute $g(x)$ into $f(x)$:
$$(f\circ g)(x) = f(g(x)) = (2x-3)^2 - 1$$
Expand and simplify:
$$(2x-3)^2 - 1 = 4x^2 -12x +9 -1 = 4x^2 -12x +8$$

Step2: Analyze domain of the function

The resulting function is a polynomial. Polynomial functions are defined for all real numbers.

Answer:

$\boldsymbol{(-\infty,\infty)}$