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 $g(x) = x -1$
3. The composition of functions is defined as $(f \circ g)(x) = f(g(x))$ and $(g \circ f)(x) = g(f(x))$

STEP 2

First, we need to find the composition $f \circ g$ . This means we need to substitute $g(x)$ into $f(x)$ .
$(f \circ g)(x) = f(g(x))$

STEP 3

Now, plug in the given function $g(x)$ into $f(x)$ .
$(f \circ g)(x) = f(x -1)$

STEP 4

Substitute $x -1$ into $f(x)$ , replacing every $x$ in $f(x)$ with $x -1$ .
$(f \circ g)(x) = (x -1)^{2}$

STEP 5

Expand the square to simplify the expression.
$(f \circ g)(x) = x^{2} -2x +1$

STEP 6

Next, we need to find the composition $g \circ f$ . This means we need to substitute $f(x)$ into $g(x)$ .
$(g \circ f)(x) = g(f(x))$

STEP 7

Now, plug in the given function $f(x)$ into $g(x)$ .
$(g \circ f)(x) = g(x^{2})$

STEP 8

Substitute $x^{2}$ into $g(x)$ , replacing every $x$ in $g(x)$ with $x^{2}$ .
$(g \circ f)(x) = x^{2} -1$

STEP 9

Finally, we need to find the composition $g \circ g$ . This means we need to substitute $g(x)$ into itself.
$(g \circ g)(x) = g(g(x))$

STEP 10

Now, plug in the given function $g(x)$ into itself.
$(g \circ g)(x) = g(x -)$

STEP 11

Substitute $x -$ into $g(x)$ , replacing every $x$ in $g(x)$ with $x -$ .
$(g \circ g)(x) = (x -) -$

STEP 12

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

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