Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7.
$ƒ (x) = \frac{2}{x}, g (x) = x + 1$

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Identify the given functions: $f(x) = \frac{2}{x}$ and $g(x) = x + 1$.

Find the composition $(f \circ g)(x)$, which means $f(g(x))$. Substitute $g(x)$ into $f$: $f(g(x)) = f(x + 1) = \frac{2}{x + 1}$.

Determine the domain of $g(x)$. Since $g(x) = x + 1$ is a polynomial, its domain is all real numbers, $(-\infty, \infty)$.

Determine the domain restrictions from $f(g(x))$. Since $f(x) = \frac{2}{x}$ is undefined when the denominator is zero, set $x + 1 \neq 0$ to avoid division by zero. This gives $x \neq -1$.

Combine the domain restrictions: the domain of $(f \circ g)(x)$ is all real numbers except $x = -1$, or $(-\infty, -1) \cup (-1, \infty)$.

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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 means you first evaluate g(x), then use that output as the input for 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 for which the function is defined. When composing functions, the domain of (f∘g)(x) includes all x-values in the domain of g such that g(x) lies in the domain of f. Identifying domain restrictions is crucial to avoid undefined expressions.

Rational Functions and Restrictions

A rational function is a ratio of two polynomials, like f(x) = 2/x, which is undefined when the denominator is zero. Recognizing these restrictions helps determine the domain by excluding values that make the denominator zero, ensuring the function remains valid.

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