Question

for f(x)=x+5 and g(x)=2x+4, find the following functions. a. (f∘g)(x); b. (g∘f)(x); c. (f∘g)(-1); d. (g∘f)(-1) 3. a. (f∘g)(x) = (simplify your answer.)

Explanation:

Step1: Define composite function

$(f \circ g)(x) = f(g(x))$

Step2: Substitute $g(x)$ into $f$

$f(g(x)) = f(2x+4)$

Step3: Apply $f$ to $2x+4$

$f(2x+4) = (2x+4) + 5 = 2x + 9$

Step4: Define $(g \circ f)(x)$

$(g \circ f)(x) = g(f(x))$

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

$g(f(x)) = g(x+5)$

Step6: Apply $g$ to $x+5$

$g(x+5) = 2(x+5) + 4 = 2x + 10 + 4 = 2x + 14$

Step7: Evaluate $(f \circ g)(-1)$

Substitute $x=-1$ into $2x+9$: $2(-1) + 9 = -2 + 9 = 7$

Step8: Evaluate $(g \circ f)(-1)$

Substitute $x=-1$ into $2x+14$: $2(-1) + 14 = -2 + 14 = 12$

Answer:

a. $2x + 9$
b. $2x + 14$
c. $7$
d. $12$