Find the inverse function of the function f, of, x, equals, minus, start fraction, 9, x, divided by, 5, end fractionf(x)=−

Question

5
9x
​
.
Answer
Attempt 1 out of 2
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, 5, x, divided by, 9, end fractionf
−1
(x)=
9
5x
​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, minus, start fraction, 5, x, divided by, 9, end fractionf
−1
(x)=−
9
5x
​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, 5, divided by, 9, x, end fractionf
−1
(x)=
9x
5
​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, minus, start fraction, 5, divided by, 9, x, end fractionf
−1
(x)=−
9x
5
​

Answer 1

To find the inverse function of \( f(x) = -\frac{9}{5}x \), we can follow these steps:

1. Replace \( f(x) \) with \( y \): \[ y = -\frac{9}{5}x \]

2. Swap \( x \) and \( y \): \[ x = -\frac{9}{5}y \]

3. Solve for \( y \): Multiply both sides by \( -\frac{5}{9} \): \[ y = -\frac{5}{9}x \]

Thus, the inverse function is: \[ f^{-1}(x) = -\frac{5}{9}x \]

Now, let's look at the multiple choice options given:

1. \( f^{-1}(x) = \frac{5x}{9} \)
2. \( f^{-1}(x) = -\frac{9}{5}x \)
3. \( f^{-1}(x) = \frac{5}{9}x \)
4. \( f^{-1}(x) = -\frac{9}{5}x \)

The correct answer based on our calculation is: \[ f^{-1}(x) = -\frac{5}{9}x \]

None of the given options exactly match this. It seems like there might be an oversight in the options provided, as the correct inverse \( -\frac{5}{9}x \) does not appear in the list.