Question

Two functions f and h are defined on the set of real numbers, R, by f(x) = (4−x^2)^1/2 and h(x) = (3−x)^1/ 2 (a) Find the domains of f and h. (b) Find foh(x).

Answer 1

a.1) Find the domain of f *

$4 - x^2 ≥ 0\iff x^2 - 4 ≤ 0 \iff \\ (x - 2)(x + 2) ≤ 0 \iff x \in [-2;2]$

a.2) Find the domain of h *

$3 - x ≥ 0 \iff x ≤ 3 \iff x \in (-\infin; 3]$

b) Find foh(x)

$f∘h(x) = f\big(h(x)\big) = f((3-x)^{1/2}) = \\=\Big(4 - \big((3-x)^{1/2}\big)^2\Big)^{1/2} = \big(4 - (3-x)\big)^{1/2} =\\= (4 - 3 + x)^{1/2} = (x + 1)^{1/2}$

*Explanation: expressions raised to the power of 1/2 must be greater than 0 or equal to 0.