Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other?
$f(x) = 2.5^x$
$g(x) = 5x + 10$

Explanation:

Step1: Analyze the types of functions

\(f(x) = 2.5^x\) is an exponential function, and \(g(x)=5x + 10\) is a linear function.

Step2: Recall the growth rates of functions

Exponential functions with a base \(> 1\) (here base \(2.5>1\)) grow faster than linear functions (of the form \(mx + b\), where \(m\) and \(b\) are constants) as \(x\) becomes very large. So we can also test some large values of \(x\) to confirm.
Let's take \(x = 5\):
\(f(5)=2.5^5=2.5\times2.5\times2.5\times2.5\times2.5 = 97.65625\)
\(g(5)=5\times5 + 10=25 + 10 = 35\)
Take \(x = 10\):
\(f(10)=2.5^{10}=(2.5^5)^2=97.65625^2\approx9536.74316\)
\(g(10)=5\times10+10 = 60\)
As \(x\) increases, the value of \(f(x)\) (exponential) becomes much larger than \(g(x)\) (linear).

Answer:

\(f(x)=2.5^x\) eventually exceeds \(g(x) = 5x+10\)