Question

which function increases at the fastest rate between x = 0 and x = 8?
linear function
f(x)=2x+2
x | f(x)
0 | 2
2 | 6
4 | 10
6 | 14
8 | 18
exponential function
f(x)=2^x + 2
x | f(x)
0 | 3
2 | 6

Explanation:

Step1: Analyze the linear function

The linear function is \( f(x) = 2x + 2 \). The rate of change (slope) of a linear function is constant. The slope \( m \) of \( f(x)=mx + b \) is 2. To find the average rate of change between \( x = 0 \) and \( x = 8 \), we can use the formula for average rate of change: \( \frac{f(8)-f(0)}{8 - 0} \).

First, find \( f(8) \) and \( f(0) \):

• \( f(0)=2(0)+2 = 2 \)
• \( f(8)=2(8)+2 = 18 \)

Then, the average rate of change is \( \frac{18 - 2}{8-0}=\frac{16}{8}=2 \).

Step2: Analyze the exponential function

The exponential function is \( f(x)=2^{x}+2 \). Let's find the average rate of change between \( x = 0 \) and \( x = 8 \).

First, find \( f(0) \) and \( f(8) \):

• \( f(0)=2^{0}+2 = 1 + 2=3 \)
• \( f(8)=2^{8}+2 = 256+2 = 258 \)

Then, the average rate of change is \( \frac{f(8)-f(0)}{8 - 0}=\frac{258 - 3}{8}=\frac{255}{8}=31.875 \).

Step3: Compare the rates of change

The average rate of change of the linear function is 2, and the average rate of change of the exponential function is 31.875. Since \( 31.875>2 \), the exponential function \( f(x)=2^{x}+2 \) increases at a faster rate between \( x = 0 \) and \( x = 8 \).

Answer:

The exponential function \( f(x) = 2^{x}+2 \) increases at the fastest rate between \( x = 0 \) and \( x = 8 \).