For $$ f(x)=x^{2}, g(x)=x-3 $$, find $$ \begin{array}{ll} ext { a. }(f \circ g)(x) & ext { b. }(g \circ f)(x)\end{array} $$

Question

For $$ f(x)=x^{2}, g(x)=x-3 $$, find $$ \begin{array}{ll}	ext { a. }(f \circ g)(x) & 	ext { b. }(g \circ f)(x)\end{array} $$

UpStudy AI · Accepted Answer

To find $$ (f \circ g)(x) $$ and $$ (g \circ f)(x) $$, we need to first understand the composition of functions.

The composition of functions $$ f $$ and $$ g $$ is denoted as $$ (f \circ g)(x) $$ and is defined as $$ f(g(x)) $$. Similarly, $$ (g \circ f)(x) $$ is defined as $$ g(f(x)) $$.

Given:
$$ f(x) = x^{2} $$
$$ g(x) = x - 3 $$

a. To find $$ (f \circ g)(x) $$:
$$ (f \circ g)(x) = f(g(x)) $$

Substitute $$ g(x) = x - 3 $$ into $$ f(x) = x^{2} $$:
$$ f(g(x)) = f(x - 3) = (x - 3)^{2} $$

Therefore, $$ (f \circ g)(x) = (x - 3)^{2} $$

b. To find $$ (g \circ f)(x) $$:
$$ (g \circ f)(x) = g(f(x)) $$

Substitute $$ f(x) = x^{2} $$ into $$ g(x) = x - 3 $$:
$$ g(f(x)) = g(x^{2}) = x^{2} - 3 $$

Therefore, $$ (g \circ f)(x) = x^{2} - 3 $$
a. $$ (f \circ g)(x) = (x - 3)^{2} $$ b. $$ (g \circ f)(x) = x^{2} - 3 $$

For \( f(x)=x^{2}, g(x)=x-3 \), find \( \begin{array}{ll}\text { a. }(f \circ g)(x) & \text { b. }(g \circ f)(x)\end{array} \)

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