Question

let $f(x)=x^2$ and $g(x)=\sqrt{x}$. 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) = $
c) $(f \circ g)(3)$
$(f \circ g)(3) = $
d) $(g \circ f)(3)$

Explanation:

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

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

Step2: Substitute $g(x)=\sqrt{x}$ into $f$

$f(g(x)) = (\sqrt{x})^2 = x$ for $x\geq0$

Step3: Define composite function $(g\circ f)(x)$

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

Step4: Substitute $f(x)=x^2$ into $g$

$g(f(x)) = \sqrt{x^2} = |x|$, or $x$ for $x\geq0$

Step5: Evaluate $(f\circ g)(3)$

Use $(f\circ g)(x)=x$, so $(f\circ g)(3)=3$

Step6: Evaluate $(g\circ f)(3)$

Use $(g\circ f)(x)=|x|$, so $(g\circ f)(3)=|3|=3$

Answer:

a) $(f \circ g)(x) = x$ (for $x\geq0$)
b) $(g \circ f)(x) = |x|$ (or $x$ for $x\geq0$)
c) $(f \circ g)(3) = 3$
d) $(g \circ f)(3) = 3$