Question

if $f(x) = 4x - 2$, what is the $y$-intercept of $f^{-1}(x)$?
options: $-2$, $\frac{1}{4}$, $-\frac{1}{2}$, $\frac{1}{2}$

Explanation:

Step1: Find the inverse function

Let \( y = f(x) = 4x - 2 \). Swap \( x \) and \( y \): \( x = 4y - 2 \). Solve for \( y \):

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So, \( f^{-1}(x) = \frac{1}{4}x + \frac{1}{2} \).

Step2: Find the y-intercept

The y-intercept of a linear function \( y = mx + b \) is \( b \) (when \( x = 0 \)). For \( f^{-1}(x) = \frac{1}{4}x + \frac{1}{2} \), when \( x = 0 \), \( y = \frac{1}{2} \).

Answer:

\(\frac{1}{2}\) (the last option, assuming the options are ordered as -2, \(\frac{1}{4}\), \(-\frac{1}{2}\), \(\frac{1}{2}\))