Question

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

Step1: Analyze function types

$f(x) = 3(2)^x$ is an exponential function (base $2>1$), and $g(x)=7x + 3$ is a linear function. Exponential functions with base $>1$ grow faster than linear functions as $x$ becomes large.

Step2: Test values for confirmation

• For $x = 1$: $f(1)=3(2)^1 = 6$, $g(1)=7(1)+3 = 10$ (here $g(x)>f(x)$).
• For $x = 2$: $f(2)=3(2)^2 = 12$, $g(2)=7(2)+3 = 17$ (still $g(x)>f(x)$).
• For $x = 3$: $f(3)=3(2)^3 = 24$, $g(3)=7(3)+3 = 24$ (equal here).
• For $x = 4$: $f(4)=3(2)^4 = 48$, $g(4)=7(4)+3 = 31$ (now $f(x)>g(x)$).
• As $x$ increases further (e.g., $x = 5$: $f(5)=3(2)^5 = 96$, $g(5)=7(5)+3 = 38$), $f(x)$ grows much faster.

Answer:

The function $f(x)=3(2)^x$ eventually exceeds $g(x)=7x + 3$.