Question

for the functions $f(x)= -6x + 9$ and $g(x)=7x + 9$ find the following
(a) $(f \circ g)(x)$ and its domain
(b) $(g \circ f)(x)$ and its domain

(a) $(f \circ g)(x)=$ (simplify your answer.)

Explanation:

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

$(f \circ g)(x) = f(g(x)) = f(7x+9)$

Step2: Replace $x$ in $f(x)$ with $7x+9$

$f(7x+9) = -6(7x+9) + 9$

Step3: Expand and simplify the expression

$=-42x - 54 + 9 = -42x - 45$

Step4: Determine the domain

Both $f(x)$ and $g(x)$ are linear functions, defined for all real numbers. So the domain is all real numbers, written as $(-\infty, \infty)$.

Step5: Substitute $f(x)$ into $g(x)$

$(g \circ f)(x) = g(f(x)) = g(-6x+9)$

Step6: Replace $x$ in $g(x)$ with $-6x+9$

$g(-6x+9) = 7(-6x+9) + 9$

Step7: Expand and simplify the expression

$=-42x + 63 + 9 = -42x + 72$

Step8: Determine the domain

Both $f(x)$ and $g(x)$ are linear functions, defined for all real numbers. So the domain is all real numbers, written as $(-\infty, \infty)$.

Answer:

(a) $(f \circ g)(x) = -42x - 45$, Domain: $(-\infty, \infty)$
(b) $(g \circ f)(x) = -42x + 72$, Domain: $(-\infty, \infty)$