Question

let $f(x) = sqrt5{3x - 1}$ and let $g(x)$ be a one - to - one function with $g^{-1}(5)=2$. if the point $(4, - 1)$ lies on the graph of $g$, find $g^{-1}(f(0))$.
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Explanation:

Step1: Calculate \( f(0) \)

We know that \( f(x)=\sqrt[5]{3x - 1} \). Substitute \( x = 0 \) into the function:
\( f(0)=\sqrt[5]{3\times0 - 1}=\sqrt[5]{- 1}=-1 \)

Step2: Find \( g^{-1}(f(0)) \) which is \( g^{-1}(-1) \)

We are given that the point \( (4,-1) \) lies on the graph of \( g \). By the definition of the inverse function, if \( (a,b) \) lies on the graph of \( g \), then \( (b,a) \) lies on the graph of \( g^{-1} \). So if \( g(4)=-1 \), then \( g^{-1}(-1) = 4 \)

Answer:

4