Question

complete the table of values for $f(x) = x^{2} + 7$ and $g(x) = 3(2)^{x}$.

$x$ $f(x)$ $g(x)$
0 $quad$ $quad$
1 $quad$ $quad$
2 $quad$ $quad$
3 $quad$ $quad$

both $f(x)$ and $g(x)$ grow as $x$ gets larger and larger. which function eventually exceeds the other?
$f(x) = x^{2} + 7$
$g(x) = 3(2)^{x}$

Explanation:

Step1: Calculate $f(x)$ at $x=0$

$f(0) = 0^2 + 7 = 7$

Step2: Calculate $g(x)$ at $x=0$

$g(0) = 3(2)^0 = 3\times1 = 3$

Step3: Calculate $f(x)$ at $x=1$

$f(1) = 1^2 + 7 = 8$

Step4: Calculate $g(x)$ at $x=1$

$g(1) = 3(2)^1 = 3\times2 = 6$

Step5: Calculate $f(x)$ at $x=2$

$f(2) = 2^2 + 7 = 4 + 7 = 11$

Step6: Calculate $g(x)$ at $x=2$

$g(2) = 3(2)^2 = 3\times4 = 12$

Step7: Calculate $f(x)$ at $x=3$

$f(3) = 3^2 + 7 = 9 + 7 = 16$

Step8: Calculate $g(x)$ at $x=3$

$g(3) = 3(2)^3 = 3\times8 = 24$

Step9: Compare long-term growth

Exponential functions grow faster than quadratic functions.

Answer:

Completed Table:

$x$	$f(x)$	$g(x)$
1	8	6
2	11	12
3	16	24

Growth Comparison:

$g(x) = 3(2)^x$