Question

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
​

Answer 1

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:
**\( f^{-1}(x) = \frac{5}{4}x \)**.