QuestionFind $f \circ g$ , $g \circ f$ , and $g \circ g$ for $f(x)=x^{2}$ and $g(x)=x-1$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as f(x) = x^. . The function $g(x)$ is defined as $g(x) = x -1$ .

STEP 2

We first need to find $f \circ g$ , which means we need to substitute $g(x)$ into $f(x)$ .
$f(g(x)) = f(x-1)$

STEP 3

Now, plug in the definition of $f(x)$ .
$f(g(x)) = (x-1)^2$

STEP 4

Expand the square.
$f(g(x)) = (x-1)^2 = x^2 -2x +1$ So, $f \circ g = x^2 -2x +1$ .

STEP 5

Next, we need to find $g \circ f$ , which means we need to substitute $f(x)$ into $g(x)$ .
$g(f(x)) = g(x^2)$

STEP 6

Now, plug in the definition of $g(x)$ .
$g(f(x)) = x^2 -1$ So, $g \circ f = x^2 -1$ .

STEP 7

Finally, we need to find $g \circ g$ , which means we need to substitute $g(x)$ into $g(x)$ again.
$g(g(x)) = g(x-1)$

STEP 8

Now, plug in the definition of $g(x)$ .
$g(g(x)) = (x-1) -1$

STEP 9

implify the expression.
$g(g(x)) = x -2$ So, $g \circ g = x -2$ .
The solutions are(a) $f \circ g = x^2 -2x +$ (b) $g \circ f = x^2 -$ (c) $g \circ g = x -2$

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