QuestionFor $f(x)=8x+3$ and $g(x)=x^{2}$ , find the composite functions: (a) $f \circ g$ , (b) $g \circ f$ , (c) $f \circ f$ , (d) $g \circ g$ and their domains.

Studdy Solution

STEP 1

Assumptions1. The functions are $f(x)=8x+3$ and $g(x)=x^{}$ . . We need to find the composite functions $f \circ g$ , $g \circ f$ , $f \circ f$ , and $g \circ g$ .
3. We also need to state the domain of each composite function.

STEP 2

First, let's find the composite function $f \circ g$ . This is done by substituting $g(x)$ into $f(x)$ .
$(f \circ g)(x) = f(g(x))$

STEP 3

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

STEP 4

Substitute $x^{2}$ into $f(x)$ .
$(f \circ g)(x) =8x^{2}+3$

STEP 5

The domain of a function is the set of all possible input values (x-values) which will produce a valid output. Since there are no restrictions on $x$ in the function $f \circ g$ , the domain is all real numbers.
$Domain\, of\, f \circ g = \{x \mid x \in \mathbb{R}\}$

STEP 6

Next, let's find the composite function $g \circ f$ . This is done by substituting $f(x)$ into $g(x)$ .
$(g \circ f)(x) = g(f(x))$

STEP 7

Now, plug in the given functions into the formula.
$(g \circ f)(x) = g(x+3)$

STEP 8

Substitute $8x+3$ into $g(x)$ .
$(g \circ f)(x) = (8x+3)^{2}$

STEP 9

The domain of the function $g \circ f$ is also all real numbers, since there are no restrictions on $x$ .
$Domain\, of\, g \circ f = \{x \mid x \in \mathbb{R}\}$

STEP 10

Next, let's find the composite function $f \circ f$ . This is done by substituting $f(x)$ into itself.
$(f \circ f)(x) = f(f(x))$

STEP 11

Now, plug in the given function into the formula.
$(f \circ f)(x) = f(8x+3)$

STEP 12

Substitute $8x+$ into $f(x)$ .
$(f \circ f)(x) =8(8x+)+$

STEP 13

implify the function.
$(f \circ f)(x) =64x+24+3 =64x+27$

STEP 14

The domain of the function $f \circ f$ is also all real numbers, since there are no restrictions on $x$ .
$Domain\, of\, f \circ f = \{x \mid x \in \mathbb{R}\}$

STEP 15

Finally, let's find the composite function $g \circ g$ . This is done by substituting $g(x)$ into itself.
$(g \circ g)(x) = g(g(x))$

STEP 16

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

STEP 17

Substitute $x^{2}$ into $g(x)$ .
$(g \circ g)(x) = (x^{2})^{2}$

STEP 18

implify the function.
$(g \circ g)(x) = x^{4}$

STEP 19

The domain of the function $g \circ g$ is also all real numbers, since there are no restrictions on $x$ .
$Domain\, of\, g \circ g = \{x \mid x \in \mathbb{R}\}$ So, the composite functions and their domains are(a) $(f \circ g)(x)=8x^{}+3$ , Domain $\{x \mid x \in \mathbb{R}\}$ (b) $(g \circ f)(x)=(8x+3)^{}$ , Domain $\{x \mid x \in \mathbb{R}\}$ (c) $(f \circ f)(x)=64x+27$ , Domain $\{x \mid x \in \mathbb{R}\}$ (d) $(g \circ g)(x)=x^{4}$ , Domain $\{x \mid x \in \mathbb{R}\}$

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