Question

x f(x) 0 1 1 4 2 16 3 64 4 256 5 1,024 x g(x) 0 6 1 11 2 16 3 21 4 26 5 31 which statement is true? a. as x increases, the rate of change of g(x) exceeds the rate of change of f(x). b. as x increases, the rate of change of f(x) exceeds the rate of change of g(x). c. at x = 2, the rate of change of g(x) is equal to the rate of change of f(x). d. on every interval of x, the rate of change of f(x) exceeds the rate of change of g(x).

Explanation:

Step1: Identify g(x) rate of change

The function \(g(x)\) has a constant difference between consecutive values: \(11-6=5\), \(16-11=5\), etc. So it is linear with a constant rate of change of \(5\).

Step2: Identify f(x) rate of change

The function \(f(x)\) is exponential: \(f(x)=4^x\). Calculate its interval rates of change:

• From \(x=0\) to \(1\): \(4-1=3\)
• From \(x=1\) to \(2\): \(16-4=12\)
• From \(x=2\) to \(3\): \(64-16=48\)
• From \(x=3\) to \(4\): \(256-64=192\)
• From \(x=4\) to \(5\): \(1024-256=768\)

Step3: Compare the rates

For early intervals, \(f(x)\) has a lower rate, but as \(x\) increases, its rate grows rapidly and surpasses \(g(x)\)'s constant rate of 5, and continues to exceed it more as \(x\) increases.

Step4: Evaluate all options

• A: False, \(g(x)\)'s rate does not exceed \(f(x)\)'s as \(x\) increases.
• C: False, at \(x=2\) (interval 1-2), \(f(x)\) rate is 12, \(g(x)\) is 5.
• D: False, first interval (0-1) has \(f(x)\) rate 3 < 5.
• B: True, as \(x\) increases, \(f(x)\)'s rate becomes and stays greater than \(g(x)\)'s.

Answer:

B. As x increases, the rate of change of f(x) exceeds the rate of change of g(x).