True or false?(a) The function f(x) = 3x + 9 has constant average rate of change on all intervals.

Step 1: Break Down the Problem

The problem asks whether the function $f(x) = 3x + 9$ has a constant average rate of change on all intervals. To determine this, we need to understand what average rate of change means for a function.

Step 2: Relevant Concepts

The average rate of change of a function $f(x)$ between two points $a$ and $b$ is given by the formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$ For a linear function like $f(x) = 3x + 9$, the rate of change is constant.

Step 3: Analysis and Detail

1. Identify the function: $f(x) = 3x + 9$
2. Calculate the average rate of change for any interval $[a, b]$: $f(a) = 3a + 9$ $f(b) = 3b + 9$ Therefore, $\text{Average Rate of Change} = \frac{(3b + 9) - (3a + 9)}{b - a} = \frac{3b - 3a}{b - a} = \frac{3(b - a)}{b - a} = 3$ The average rate of change simplifies to 3, which is constant regardless of the interval $[a, b]$.

Step 4: Verify and Summarize

Since the average rate of change for the function $f(x) = 3x + 9$ is consistently 3, we can conclude that it exhibits a constant rate of change on all intervals.

Final Answer

True: The function $f(x) = 3x + 9$ has a constant average rate of change on all intervals.