Question

let $f(x)=\sqrt{x+1}$ and $g(x)=2x-3$. 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 $f$

Substitute $g(x)=2x-3$ into $f(t)=\sqrt{t+1}$:
$(f \circ g)(x) = \sqrt{(2x-3) + 1}$

Step3: Simplify the radicand

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

Step4: Define composite function $g\circ f$

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

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

Substitute $f(x)=\sqrt{x+1}$ into $g(t)=2t-3$:
$(g \circ f)(x) = 2\sqrt{x+1} - 3$

Answer:

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