Question

1. the function $f(x) = 10(1.05)^x$ models the population of a species after $x$ years. does this function represent exponential growth or decay? show your work here

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( f(x) = a(b)^x \), where \( a \) is the initial amount and \( b \) is the base.

Step2: Analyze the base value

In the function \( f(x) = 10(1.05)^x \), the base \( b = 1.05 \).
For exponential growth, the base \( b>1 \), and for exponential decay, the base \( 0 < b < 1 \).
Since \( 1.05>1 \), as \( x \) (number of years) increases, the value of \( (1.05)^x \) will increase, and thus the population \( f(x) \) will increase over time.

Answer:

This function represents exponential growth.