Find a. (f ○ g)(x); b. the domain of (f ○ g). f(x) = (x + 1)/(x - 2), g(x) = 1/x

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Step 1: Understand that the composition (f \(\circ\) g)(x) means f(g(x)), which is the function f applied to the output of g(x). So, first substitute g(x) into f(x).

Step 2: Write the expression for (f \(\circ\) g)(x) by replacing every x in f(x) with g(x). Since f(x) = \(\frac{x + 1}{x - 2}\) and g(x) = \(\frac{1}{x}\), we have (f \(\circ\) g)(x) = f\(\left\)(\(\frac{1}{x}\)\(\right\)) = \(\frac{\frac{1}{x}\) + 1}{\(\frac{1}{x}\) - 2}.

Step 3: Simplify the complex fraction by finding a common denominator for the numerator and denominator separately, then simplify the overall expression.

Step 4: Determine the domain of (f \(\circ\) g)(x) by considering the domain restrictions of both g(x) and f(g(x)). First, find where g(x) is defined (denominator \(\neq\) 0), then find where f is defined when its input is g(x) (denominator of f(g(x)) \(\neq\) 0).

Step 5: Combine the domain restrictions from both functions to write the domain of (f \(\circ\) g)(x) in interval or set notation, excluding values that make any denominator zero or cause undefined expressions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Composition

Function composition involves applying one function to the result of another, denoted as (f ○ g)(x) = f(g(x)). It requires substituting the entire output of g(x) into the function f. Understanding this process is essential to correctly find (f ○ g)(x).

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When composing functions, the domain of (f ○ g) includes all x-values in the domain of g for which g(x) is in the domain of f. Identifying restrictions like division by zero is crucial.

Rational Functions and Restrictions

Rational functions are ratios of polynomials, which can have restrictions where the denominator equals zero. For f(x) = (x + 1)/(x - 2) and g(x) = 1/x, values that make denominators zero must be excluded from the domain to avoid undefined expressions.

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