Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=1/(x-2), g(x)=1/x

Verified step by step guidance

1. Identify the composition of functions: \((g \circ f)(x) = g(f(x))\).
2. Substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = g\left(\frac{1}{x-2}\right)\).
3. Evaluate \(g\left(\frac{1}{x-2}\right)\): Since \(g(x) = \frac{1}{x}\), substitute to get \(\frac{1}{\frac{1}{x-2}}\).
4. Simplify the expression: \(\frac{1}{\frac{1}{x-2}} = x-2\).
5. Determine the domain: Exclude values that make the original functions undefined, i.e., \(x \neq 2\) and \(x \neq 0\).

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, (g∘ƒ)(x) means applying function f first and then applying function g to the result. Understanding how to correctly perform this operation is crucial for analyzing the behavior of the composed function.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the functions f(x) = 1/(x-2) and g(x) = 1/x, it is essential to identify values that would make the denominator zero, as these values are excluded from the domain. This understanding is vital for determining the domain of the composed function (g∘ƒ)(x).

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the function approaches infinity as the input approaches a certain value. For the functions given, vertical asymptotes can be found at the values that make the denominator zero. Recognizing these asymptotes helps in understanding the behavior of the functions and their compositions, particularly in relation to their domains.