Question

find f(g(x)) and g(f(x)). state the domain of each.
f(x) = 4x + 9; g(x) = x - 8
what is the formula for f(g(x))?
f(g(x)) = \square (simplify your answer.)
what is the domain of f(g(x))?
\square (type your answer in interval notation.)
what is the formula for g(f(x))?
g(f(x)) = \square (simplify your answer.)
what is the domain of g(f(x))?
\square (type your answer in interval notation.)

Explanation:

Step1: Substitute $g(x)$ into $f(x)$

$f(g(x)) = 4(x-8) + 9$

Step2: Simplify the expression

$f(g(x)) = 4x - 32 + 9 = 4x - 23$

Step3: Determine domain of $f(g(x))$

The function $4x-23$ is linear; all real numbers are allowed. Domain: $(-\infty, \infty)$

Step4: Substitute $f(x)$ into $g(x)$

$g(f(x)) = (4x+9) - 8$

Step5: Simplify the expression

$g(f(x)) = 4x + 9 - 8 = 4x + 1$

Step6: Determine domain of $g(f(x))$

The function $4x+1$ is linear; all real numbers are allowed. Domain: $(-\infty, \infty)$

Answer:

Formula for $f(g(x))$: $4x - 23$
Domain of $f(g(x))$: $(-\infty, \infty)$
Formula for $g(f(x))$: $4x + 1$
Domain of $g(f(x))$: $(-\infty, \infty)$