Question
3. Given \( f(x)=(x+3)(x-5) \), determine the \( x \)-intercepts fot \( f\left(\frac{1}{3} x\right) \)
Ask by Murphy Murray. in Canada
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The x-intercepts are at x = –9 and x = 15.
Solution
To find the \( x \)-intercepts of the function \( f\left(\frac{1}{3} x\right) \), follow these steps:
1. **Start with the given function:**
\[
f(x) = (x + 3)(x - 5)
\]
2. **Substitute \( \frac{1}{3} x \) into the function:**
\[
f\left(\frac{1}{3} x\right) = \left(\frac{1}{3} x + 3\right) \left(\frac{1}{3} x - 5\right)
\]
3. **Set the function equal to zero to find the \( x \)-intercepts:**
\[
\left(\frac{1}{3} x + 3\right) \left(\frac{1}{3} x - 5\right) = 0
\]
4. **Solve each factor for \( x \):**
- **First factor:**
\[
\frac{1}{3} x + 3 = 0 \\
\frac{1}{3} x = -3 \\
x = -9
\]
- **Second factor:**
\[
\frac{1}{3} x - 5 = 0 \\
\frac{1}{3} x = 5 \\
x = 15
\]
5. **Conclusion:**
The \( x \)-intercepts of \( f\left(\frac{1}{3} x\right) \) are at:
\[
(-9, 0) \quad \text{and} \quad (15, 0)
\]
**Final Answer:**
The x-intercepts are at x = –9 and x = 15.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To find the \( x \)-intercepts of \( f\left(\frac{1}{3} x\right) \), start by substituting \( \frac{1}{3} x \) into the function \( f(x) \): \[ f\left(\frac{1}{3} x\right) = \left(\frac{1}{3} x + 3\right)\left(\frac{1}{3} x - 5\right) \] Set \( f\left(\frac{1}{3} x\right) = 0 \) to find the \( x \)-intercepts: \[ \left(\frac{1}{3} x + 3\right)\left(\frac{1}{3} x - 5\right) = 0 \] This gives us two equations: 1. \( \frac{1}{3} x + 3 = 0 \) leads to \( \frac{1}{3} x = -3 \) or \( x = -9 \). 2. \( \frac{1}{3} x - 5 = 0 \) leads to \( \frac{1}{3} x = 5 \) or \( x = 15 \). Thus, the \( x \)-intercepts of \( f\left(\frac{1}{3} x\right) \) are \( x = -9 \) and \( x = 15 \). Happy graphing!
