Question

complete the sentences about the functions $f(x) = 6x + 9$ and $g(x) = 3(3)^x$.
for small values of $x$, the function with greater values is . for instance, whereas . as the value of $x$ increases, however, eventually has greater values.

Explanation:

Step1: Test small x (e.g., x=0)

Calculate $f(0)=6(0)+9=9$, $g(0)=3(3)^0=3(1)=3$

Step2: Compare small x values

$f(0)=9 > g(0)=3$, so $f(x)$ is greater for small x.

Step3: Test larger x (e.g., x=3)

Calculate $f(3)=6(3)+9=18+9=27$, $g(3)=3(3)^3=3(27)=81$

Step4: Compare large x values

$g(3)=81 > f(3)=27$, so $g(x)$ overtakes as x increases.

Answer:

For small values of $x$, the function with greater values is $\boldsymbol{f(x) = 6x + 9}$. For instance, $\boldsymbol{f(0) = 9}$ whereas $\boldsymbol{g(0) = 3}$. As the value of $x$ increases, however, $\boldsymbol{g(x) = 3(3)^x}$ eventually has greater values.