Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.ƒ(x)=√x,g(x)=1(x+5)ƒ(x)=\surd x,g(x)=\frac{1}{(x+5)}

Ch. 2 - Graphs and Functions

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Ch. 2 - Graphs and Functions

문제 85b

3장, 문제 85b

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.
$ƒ (x) = \sqrt x, g (x) = \frac{1}{(x + 5)}$

검증된 단계별 안내

First, understand that the composition of functions (g $\circ$ f)(x) means you substitute the function f(x) into g(x). In other words, (g $\circ$ f)(x) = g(f(x)).

Given f(x) = $\sqrt{x}$ and g(x) = $\frac{1}{x+5}$, substitute f(x) into g(x) to get (g $\circ$ f)(x) = $\frac{1}{\sqrt{x}$ + 5}.

Next, determine the domain of (g $\circ$ f)(x). Start by considering the domain of f(x) = $\sqrt{x}$, which requires that the expression inside the square root is non-negative, so x $\geq$ 0.

Then, consider the domain restrictions from g(x) when applied to f(x). Since g(x) = $\frac{1}{x+5}$, the denominator cannot be zero. So, set $\sqrt{x}$ + 5 $\neq$ 0 and solve for x.

Combine the domain restrictions from both functions to find the overall domain of (g $\circ$ f)(x). This means x must satisfy both x $\geq$ 0 and $\sqrt{x}$ + 5 $\neq$ 0.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Function Composition

Function composition involves applying one function to the result of another, denoted as (g∘f)(x) = g(f(x)). It requires substituting the entire output of f(x) into g(x), creating a new function that combines both operations.

Domain of a Function

The domain is the set of all input values for which a function is defined. When composing functions, the domain of (g∘f)(x) includes all x-values in the domain of f for which f(x) is in the domain of g.

Square Root and Rational Function Domains

The square root function ƒ(x) = √x requires x ≥ 0 to avoid imaginary numbers. The rational function g(x) = 1/(x+5) is undefined when the denominator is zero, so x ≠ -5. These restrictions affect the domain of the composite function.

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.ƒ(x)=√x,g(x)=1(x+5)ƒ(x)=\(\surd\) x,g(x)=\(\frac{1}{(x+5)}\)