Solved on Sep 12, 2023

Find composite functions and their domains for f(x)=xf(x)=\sqrt{x} and g(x)=8x+9g(x)=8x+9. (a) fgf \circ g: 8x+9\sqrt{8x+9}, domain x98x \geq -\frac{9}{8}. (b) gfg \circ f: 8x+98\sqrt{x}+9.

STEP 1

Assumptions1. The functions are f(x)=xf(x)=\sqrt{x} and g(x)=8x+9g(x)=8x+9 . We need to find the composite functions fgf \circ g, gfg \circ f, fff \circ f, and ggg \circ g
3. We also need to state the domain of each composite function

STEP 2

First, let's find the composite function fgf \circ g. This is done by replacing every instance of xx in f(x)f(x) with g(x)g(x).
(fg)(x)=f(g(x))(f \circ g)(x) = f(g(x))

STEP 3

Now, plug in the given functions f(x)f(x) and g(x)g(x) to find fgf \circ g.
(fg)(x)=f(8x+9)(f \circ g)(x) = f(8x+9)

STEP 4

Substitute 8x+98x+9 into f(x)f(x).
(fg)(x)=8x+9(f \circ g)(x) = \sqrt{8x+9}

STEP 5

Next, let's find the domain of fgf \circ g. The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For a square root function, the value inside the square root (the radicand) must be greater than or equal to0.
8x+908x+9 \geq0

STEP 6

olve the inequality to find the domain of fgf \circ g.
x98x \geq -\frac{9}{8}So, the domain of fgf \circ g is {xx98}\left\{x \mid x \geq-\frac{9}{8}\right\}.

STEP 7

Now, let's find the composite function gfg \circ f. This is done by replacing every instance of xx in g(x)g(x) with f(x)f(x).
(gf)(x)=g(f(x))(g \circ f)(x) = g(f(x))

STEP 8

Now, plug in the given functions f(x)f(x) and g(x)g(x) to find gfg \circ f.
(gf)(x)=g(x)(g \circ f)(x) = g(\sqrt{x})

STEP 9

Substitute x\sqrt{x} into g(x)g(x).
(gf)(x)=8x+9(g \circ f)(x) =8\sqrt{x}+9

STEP 10

Next, let's find the domain of gfg \circ f. The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For a square root function, the value inside the square root (the radicand) must be greater than or equal to0.
x0x \geq0So, the domain of gfg \circ f is {xx0}\left\{x \mid x \geq0\right\}.

STEP 11

Now, let's find the composite function fff \circ f. This is done by replacing every instance of xx in f(x)f(x) with f(x)f(x).
(ff)(x)=f(f(x))(f \circ f)(x) = f(f(x))

STEP 12

Now, plug in the given function f(x)f(x) to find fff \circ f.
(ff)(x)=f(x)(f \circ f)(x) = f(\sqrt{x})

STEP 13

Substitute x\sqrt{x} into f(x)f(x).
(ff)(x)=x(f \circ f)(x) = \sqrt{\sqrt{x}}

STEP 14

Next, let's find the domain of fff \circ f. The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For a square root function, the value inside the square root (the radicand) must be greater than or equal to0.
x0x \geq0So, the domain of fff \circ f is {xx0}\left\{x \mid x \geq0\right\}.

STEP 15

Finally, let's find the composite function ggg \circ g. This is done by replacing every instance of xx in g(x)g(x) with g(x)g(x).
(gg)(x)=g(g(x))(g \circ g)(x) = g(g(x))

STEP 16

Now, plug in the given function g(x)g(x) to find ggg \circ g.
(gg)(x)=g(8x+9)(g \circ g)(x) = g(8x+9)

STEP 17

Substitute x+9x+9 into g(x)g(x).
(gg)(x)=(x+9)+9(g \circ g)(x) =(x+9)+9

STEP 18

implify the expression to find ggg \circ g.
(gg)(x)=64x+72+=64x+81(g \circ g)(x) =64x+72+ =64x+81

STEP 19

The domain of ggg \circ g is all real numbers since the function ggg \circ g is a linear function and there are no restrictions on the input values.
So, the solutions are(a) (fg)(x)=8x+9(f \circ g)(x)=\sqrt{8x+9} with domain {xx98}\left\{x \mid x \geq-\frac{9}{8}\right\} (b) (gf)(x)=8x+9(g \circ f)(x)=8\sqrt{x}+9 with domain {xx}\left\{x \mid x \geq\right\} (c) (ff)(x)=x(f \circ f)(x)=\sqrt{\sqrt{x}} with domain {xx}\left\{x \mid x \geq\right\} (d) (gg)(x)=64x+81(g \circ g)(x)=64x+81 with domain all real numbers.

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