Question
nclude " \( f^{-1}(x)= \) " in your answer. \( \quad f(x)=\frac{5 x-8}{9 x+6} \)
Ask by Nguyen Washington. in the United States
Mar 01,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f^{-1}(x) = \frac{-6x - 8}{9x - 5} \)
Solution
To find the inverse of f(x) = (5x – 8)/(9x + 6), start by writing y = (5x – 8)/(9x + 6) and then solve for x in terms of y.
Step 1: Write the equation with y:
y = (5x – 8)/(9x + 6)
Step 2: Multiply both sides by (9x + 6) to eliminate the fraction:
y(9x + 6) = 5x – 8
Step 3: Distribute y on the left side:
9xy + 6y = 5x – 8
Step 4: Gather all terms involving x on one side:
9xy – 5x = –8 – 6y
Step 5: Factor out x on the left side:
x(9y – 5) = –8 – 6y
Step 6: Solve for x by dividing by (9y – 5):
x = (–8 – 6y)/(9y – 5)
Since x was expressed in terms of y, we now switch the variables (replace y with x) to get the inverse function:
f⁻¹(x) = (–8 – 6x)/(9x – 5)
Thus, the inverse function is:
f⁻¹(x) = (–6x – 8)/(9x – 5)
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To find the inverse function \( f^{-1}(x) \), we start with \( y = f(x) = \frac{5x - 8}{9x + 6} \). To solve for \( x \), we can interchange \( x \) and \( y \) and solve for \( y \): 1. \( x = \frac{5y - 8}{9y + 6} \) 2. Multiply both sides by \( 9y + 6 \): \( x(9y + 6) = 5y - 8 \) 3. Rearrange it: \( 9xy + 6x = 5y - 8 \) 4. Collect all \( y \) terms on one side: \( 9xy - 5y = -6x - 8 \) 5. Factor out \( y \): \( y(9x - 5) = -6x - 8 \) 6. Thus, \( y = \frac{-6x - 8}{9x - 5} \) Therefore, \( f^{-1}(x) = \frac{-6x - 8}{9x - 5} \). To verify, we can calculate \( f(f^{-1}(x)) \) and check if we return to \( x \). It’s baffling how functions can flip on their heads, isn’t it?
