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

검증된 단계별 안내

Recall that the composition of functions (g $\circ$ f)(x) means g(f(x)), which is the function g applied to the output of f(x).

Start by substituting f(x) into g(x). Since f(x) = $\frac{2}{x}$, replace the input of g with $\frac{2}{x}$, so (g $\circ$ f)(x) = g$\left$($\frac{2}{x}$$\right$).

Now, apply the function g to $\frac{2}{x}$. Since g(x) = x + 1, replace x in g(x) with $\frac{2}{x}$, giving (g $\circ$ f)(x) = $\frac{2}{x}$ + 1.

Next, determine the domain of (g $\circ$ f)(x). The domain consists of all x-values for which f(x) is defined and for which g(f(x)) is defined. Since f(x) = $\frac{2}{x}$, x cannot be zero (division by zero is undefined). Also, check if g has any restrictions on its input; since g(x) = x + 1 is defined for all real numbers, no further restrictions come from g.

Therefore, the domain of (g $\circ$ f)(x) is all real numbers except x = 0.

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주요 개념

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Function Composition

Function composition involves applying one function to the result of another, denoted as (g∘f)(x) = g(f(x)). It means you first evaluate f at x, then use that output as the input for g. Understanding this process is essential to correctly find (g∘f)(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 (g∘f) includes all x-values in the domain of f for which f(x) is in the domain of g. Identifying domain restrictions is crucial to avoid undefined expressions.

Rational Functions and Their Restrictions

A rational function is a ratio of two polynomials, like f(x) = 2/x, which is undefined where the denominator is zero. Recognizing these restrictions helps determine the domain of f and, consequently, the domain of the composition (g∘f).

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