Question

compare ( f(x) = x + 2 ) shown in the table below and ( g(x) = 2^x ) shown in the graph. which function has the greater ( y )-intercept, and for which positive values of ( x ) is ( g(x) ) greater than ( f(x) )?

choose the correct words or phrases to complete the statements.

• the ( y )-intercept of ( f(x) ) is (\boldsymbol{\text{select choice}}) the intercept of ( g(x) ).
• ( g(x) ) is greater than ( f(x) ) when ( x ) is (\boldsymbol{\text{select choice}}).

Explanation:

Step1: Find y-intercept of $f(x)$

For $f(x)=x+2$, set $x=0$:
$f(0)=0+2=2$

Step2: Find y-intercept of $g(x)$

For $g(x)=2^x$, set $x=0$:
$g(0)=2^0=1$

Step3: Compare y-intercepts

$2 > 1$, so $f(x)$'s y-intercept is larger.

Step4: Find intersection of $f(x)$ and $g(x)$

Solve $x+2=2^x$. Testing values:

• $x=2$: $f(2)=4$, $g(2)=4$ (intersection point)
• $x=3$: $f(3)=5$, $g(3)=8$, so $g(x)>f(x)$ when $x>2$

Answer:

• The y-intercept of $f(x)$ is greater than the intercept of $g(x)$.
• $g(x)$ is greater than $f(x)$ when $x$ is greater than 2