Question
A function is given. Determine the average rate of change of the function between the \( g \) i \[ g(x)=\frac{9}{x+3} ; \quad x=0, x=h \]
Ask by May Medina. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The average rate of change is \(-\frac{3}{h + 3}\).
Solution
To determine the **average rate of change** of the function \( g(x) = \frac{9}{x+3} \) between \( x = 0 \) and \( x = h \), we use the formula:
\[
\text{Average Rate of Change} = \frac{g(h) - g(0)}{h - 0}
\]
Let's compute each part step-by-step:
1. **Evaluate \( g(0) \) and \( g(h) \):**
\[
g(0) = \frac{9}{0 + 3} = \frac{9}{3} = 3
\]
\[
g(h) = \frac{9}{h + 3}
\]
2. **Substitute these into the average rate of change formula:**
\[
\text{Average Rate of Change} = \frac{\frac{9}{h + 3} - 3}{h}
\]
3. **Simplify the expression:**
\[
\frac{\frac{9}{h + 3} - 3}{h} = \frac{9 - 3(h + 3)}{h(h + 3)} = \frac{9 - 3h - 9}{h(h + 3)} = \frac{-3h}{h(h + 3)}
\]
4. **Cancel out the common terms:**
\[
\frac{-3h}{h(h + 3)} = \frac{-3}{h + 3}
\]
**Final Answer:**
The average rate of change is –3 divided by (h plus three). Thus,
Average Rate = − 3 ⁄ (h + 3)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To find the average rate of change of the function \( g(x) = \frac{9}{x+3} \) between \( x=0 \) and \( x=h \), we use the formula: \[ \text{Average Rate of Change} = \frac{g(h) - g(0)}{h - 0} \] First, calculate \( g(0) \): \[ g(0) = \frac{9}{0 + 3} = \frac{9}{3} = 3 \] Next, calculate \( g(h) \): \[ g(h) = \frac{9}{h + 3} \] Now substitute these results into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{\frac{9}{h+3} - 3}{h} \] Simplifying this expression will give you your final answer! To have fun with this, consider that as \( h \) approaches infinity, the rate of change approaches zero—like watching a tortoise slowly crawl away!
