Question

express the function \\(\frac{1}{x - 1}\\) in the form \\(f \circ g\\). if \\(g(x) = x - 1\\), find the function \\(f(x)\\). your answer is \\(f(x) = \\) , next item

Explanation:

Step1: Recall the composition of functions

The composition of functions \( f \circ g \) is defined as \( (f \circ g)(x)=f(g(x)) \). We know that \( (f \circ g)(x)=\frac{1}{x - 1} \) and \( g(x)=x - 1 \). Let \( y = g(x)=x - 1 \), then \( (f \circ g)(x)=f(y)=\frac{1}{y} \).

Step2: Determine the function \( f \)

Since \( y = g(x) \), we can replace \( y \) with \( x \) (because the variable name is just a placeholder) to get the function \( f \). So if \( f(y)=\frac{1}{y} \), then \( f(x)=\frac{1}{x} \).

Answer:

\( f(x)=\frac{1}{x} \)