Question
37. \( f(x)=\frac{4}{3} x-9 \), what is. \( f^{-1}(-3) \) ? a. -13 b. -9.5 c. -7 d. 4.5
Ask by Warner Bowen. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( f^{-1}(-3) = 4.5 \)
Solution
To find \( f^{-1}(-3) \) for the function \( f(x) = \frac{4}{3}x - 9 \), we need to determine the value of \( x \) such that \( f(x) = -3 \).
**Step 1: Set up the equation**
\[
\frac{4}{3}x - 9 = -3
\]
**Step 2: Solve for \( x \)**
1. Add 9 to both sides:
\[
\frac{4}{3}x = -3 + 9
\]
\[
frac{4}{3}x = 6
\]
2. Multiply both sides by \( \frac{3}{4} \) to isolate \( x \):
\[
x = 6 \times \frac{3}{4}
\]
\[
x = \frac{18}{4}
\]
\[
x = 4.5
\]
**Conclusion:**
\( f^{-1}(-3) = 4.5 \)
**Answer:**
d. 4.5
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find \( f^{-1}(-3) \), we first need to determine the inverse function \( f^{-1}(x) \). Start with \( y = f(x) = \frac{4}{3} x - 9 \) and then solve for \( x \): 1. Rearranging gives \( y + 9 = \frac{4}{3} x \). 2. Multiplying both sides by \(\frac{3}{4}\), we get \( x = \frac{3}{4}(y + 9) \). Thus, the inverse function is \( f^{-1}(x) = \frac{3}{4}(x + 9) \). Now, substituting \(-3\) into the inverse function: \[ f^{-1}(-3) = \frac{3}{4}(-3 + 9) = \frac{3}{4} \cdot 6 = \frac{18}{4} = 4.5. \] So, the answer is \( \boxed{4.5} \).
