Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other?
$f(x) = 13x$
$g(x) = 3^x$

Explanation:

Step1: Analyze the types of functions

$f(x) = 13x$ is a linear function, and its growth rate is constant (the slope is 13). $g(x)=3^x$ is an exponential function with a base of 3 (where the base $3>1$).

Step2: Recall the growth rates of linear and exponential functions

For a linear function $y = mx + b$ (in this case, $b = 0$ and $m = 13$), the rate of change is constant. For an exponential function $y = a^x$ where $a>1$, the growth rate increases as $x$ increases. Exponential functions with $a > 1$ eventually outpace linear functions as $x$ becomes sufficiently large.

We can also test some large values of $x$:

• When $x = 1$: $f(1)=13\times1 = 13$, $g(1)=3^1=3$, so $f(x)>g(x)$ here.
• When $x = 2$: $f(2)=13\times2 = 26$, $g(2)=3^2 = 9$, so $f(x)>g(x)$ here.
• When $x = 3$: $f(3)=13\times3=39$, $g(3)=3^3 = 27$, so $f(x)>g(x)$ here.
• When $x = 4$: $f(4)=13\times4 = 52$, $g(4)=3^4=81$, now $g(x)>f(x)$.
• As $x$ increases further (e.g., $x = 5$: $f(5)=65$, $g(5)=243$; $x = 10$: $f(10)=130$, $g(10)=59049$), the exponential function $g(x)=3^x$ grows much faster than the linear function $f(x)=13x$.

Answer:

The function $g(x) = 3^x$ eventually exceeds the function $f(x)=13x$.