Question

complete the table of values for $f(x) = 3^{x + 1}$ and $g(x) = x^2 + 5$.

$x$	$f(x)$	$g(x)$
1	\boxed{}	\boxed{}
2	\boxed{}	\boxed{}
3	\boxed{}	\boxed{}

both $f(x)$ and $g(x)$ grow as $x$ gets larger and larger. which function eventually exceeds the other?
$f(x) = 3^{x + 1}$ \quad $g(x) = x^2 + 5$

Explanation:

Step1: Calculate $f(0)$ and $g(0)$

For $f(0)$: $f(0)=3^{0+1}=3^1=3$
For $g(0)$: $g(0)=0^2+5=0+5=5$

Step2: Calculate $f(1)$ and $g(1)$

For $f(1)$: $f(1)=3^{1+1}=3^2=9$
For $g(1)$: $g(1)=1^2+5=1+5=6$

Step3: Calculate $f(2)$ and $g(2)$

For $f(2)$: $f(2)=3^{2+1}=3^3=27$
For $g(2)$: $g(2)=2^2+5=4+5=9$

Step4: Calculate $f(3)$ and $g(3)$

For $f(3)$: $f(3)=3^{3+1}=3^4=81$
For $g(3)$: $g(3)=3^2+5=9+5=14$

Step5: Compare long-term growth

Exponential functions grow faster than quadratic functions as $x$ increases.

Answer:

Completed Table:

$x$	$f(x)$	$g(x)$
1	9	6
2	27	9
3	81	14

Growth Comparison Answer:

$f(x) = 3^{x+1}$