Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other?
$f(x) = \frac{1}{2}(5)^x$
$g(x) = 6x + 2$

Explanation:

Step1: Analyze the types of functions

$f(x)=\frac{1}{2}(5)^x$ is an exponential function with base $5>1$, and $g(x) = 6x + 2$ is a linear function.

Step2: Recall the growth rates of exponential and linear functions

Exponential functions with base $> 1$ grow much faster than linear functions as $x$ becomes very large. For example, when $x = 3$:

• $f(3)=\frac{1}{2}(5)^3=\frac{1}{2}\times125 = 62.5$
• $g(3)=6\times3 + 2=18 + 2 = 20$

When $x = 4$:

• $f(4)=\frac{1}{2}(5)^4=\frac{1}{2}\times625 = 312.5$
• $g(4)=6\times4 + 2=24 + 2 = 26$

As $x$ increases further, the exponential function $f(x)$ will grow much more rapidly than the linear function $g(x)$ because the exponent in the exponential function causes the value to multiply repeatedly, while the linear function only adds a constant amount each time.

Answer:

The function $f(x)=\frac{1}{2}(5)^x$ (or $f(x)$) eventually exceeds $g(x)=6x + 2$ (or $g(x)$).