Question

if $f(x) = 4x + 3$ and $g(x) = 2x - 1$, to find $(f \circ g)(x)$ the correct substitution is
○ $4(2x-1) + 3$
○ $2(4x+3) - 1$
○ $(2x-1) + 3$
○ $(4x+3) - 1$
question 6 2 pts
if $f(x) = x + 3$, $g(x) = 2x$, and $h(x) = x^2$
to find $h(f(x))$
○ $2x^2$
○ $x^2 + 9$
○ $x^2+6x+9$
○ $x^2 +3$

Explanation:

Step1: Define function composition

For $(f \circ g)(x) = f(g(x))$, substitute $g(x)$ into $f(x)$.
Given $f(x)=4x+3$, $g(x)=2x-1$, replace $x$ in $f(x)$ with $2x-1$:
$\boldsymbol{f(g(x))=4(2x-1)+3}$

Step2: Define function composition for $h(f(x))$

For $h(f(x))$, substitute $f(x)$ into $h(x)$.
Given $h(x)=x^2$, $f(x)=x+3$, replace $x$ in $h(x)$ with $x+3$:
$\boldsymbol{h(f(x))=(x+3)^2}$

Step3: Expand the squared expression

Use $(a+b)^2=a^2+2ab+b^2$ where $a=x$, $b=3$:
$(x+3)^2 = x^2 + 2(x)(3) + 3^2 = x^2+6x+9$

Answer:

1. 4(2x-1) + 3
2. $x^2+6x+9$