Find the inverse function of the function f, of, x, equals, start fraction, 4, x, divided by, 5, end fractionf(x)=

5
4x

.
Answer
Attempt 1 out of 2
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, minus, start fraction, 5, x, divided by, 4, end fractionf
−1
(x)=−
4
5x


f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, 4, x, divided by, 5, end fractionf
−1
(x)=
5
4x


f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, 5, x, divided by, 4, end fractionf
−1
(x)=
4
5x


f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, minus, start fraction, 4, x, divided by, 5, end fractionf
−1
(x)=−
5
4x

1 answer

To find the inverse function of \( f(x) = \frac{4}{5}x \), we will follow these steps:

  1. Set \( f(x) \) equal to \( y \): \[ y = \frac{4}{5}x \]

  2. Swap \( x \) and \( y \): \[ x = \frac{4}{5}y \]

  3. Solve for \( y \): To isolate \( y \), first multiply both sides by \( \frac{5}{4} \): \[ y = \frac{5}{4}x \]

  4. Write the inverse function: Thus, the inverse function \( f^{-1}(x) \) is: \[ f^{-1}(x) = \frac{5}{4}x \]

Now let's analyze the multiple-choice options:

  • \( f^{-1}(x) = -\frac{5}{4}x \)
  • \( f^{-1}(x) = \frac{4}{5}x \)
  • \( f^{-1}(x) = \frac{5}{4}x \)
  • \( f^{-1}(x) = -\frac{4}{5}x \)

The correct inverse function is: \[ f^{-1}(x) = \frac{5}{4}x \]

So, the answer is: \( f^{-1}(x) = \frac{5}{4}x \).