Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other?
$f(x) = 5x + 22$
$g(x) = 2^x$

Explanation:

Step1: Analyze the types of functions

$f(x) = 5x + 22$ is a linear function (degree 1), and $g(x)=2^x$ is an exponential function (base 2, exponent $x$).

Step2: Recall the growth rates of functions

Linear functions have a constant rate of change (slope), while exponential functions with base $> 1$ grow faster than linear functions as $x$ becomes large. This is because the exponent in the exponential function causes the output to increase multiplicatively, while the linear function increases additively.

Step3: Test with large values of $x$ (optional but illustrative)

For example, let $x = 10$:

• $f(10)=5(10)+22 = 50 + 22 = 72$
• $g(10)=2^{10}=1024$

For $x = 20$:

• $f(20)=5(20)+22 = 100 + 22 = 122$
• $g(20)=2^{20}=1048576$

We can see that as $x$ increases, $g(x)$ (the exponential function) grows much faster and will eventually exceed $f(x)$ (the linear function).

Answer:

The function $g(x) = 2^x$ eventually exceeds $f(x) = 5x + 22$.