Question

let $f(x) = \sqrt{x + 1}$ and $g(x) = 3x - 3$. find $f \circ g$ and $g \circ f$.
a) $(f \circ g)(x)$
$(f \circ g)(x) = $
b) $(g \circ f)(x)$
$(g \circ f)(x) = $

Explanation:

Step1: Define composite function $f\circ g$

$(f \circ g)(x) = f(g(x))$

Step2: Substitute $g(x)$ into $f$

$f(g(x)) = \sqrt{(3x-3) + 1}$

Step3: Simplify the radicand

$\sqrt{3x - 3 + 1} = \sqrt{3x - 2}$

Step4: Define composite function $g\circ f$

$(g \circ f)(x) = g(f(x))$

Step5: Substitute $f(x)$ into $g$

$g(f(x)) = 3(\sqrt{x+1}) - 3$

Answer:

a) $(f \circ g)(x) = \sqrt{3x - 2}$
b) $(g \circ f)(x) = 3\sqrt{x+1} - 3$