Given functions f and g, find (a)(ƒ∘g)(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 85a

3장, 문제 85a

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

검증된 단계별 안내

Identify the given functions: $f(x) = \sqrt{x}$ and $g(x) = \frac{1}{x+5}$.

Find the composition $(f \circ g)(x)$, which means substituting $g(x)$ into $f$. So, write $(f \circ g)(x) = f(g(x)) = \sqrt{g(x)}$.

Substitute $g(x)$ into $f$: $(f \circ g)(x) = \sqrt{\frac{1}{x+5}}$.

Determine the domain of $(f \circ g)(x)$ by considering the restrictions from both $f$ and $g$. Since $f(x) = \sqrt{x}$ requires the input to be $\geq 0$, set the inside of the square root $\frac{1}{x+5} \geq 0$.

Solve the inequality $\frac{1}{x+5} \geq 0$ to find the domain of $(f \circ g)(x)$. Also, exclude any values that make the denominator zero, i.e., $x \neq -5$.

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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 (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 form the composite function.

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) depends on the domain of g and the values of g(x) that lie within the domain of f. Identifying these restrictions ensures the composite function is valid.

Square Root and Rational Function Restrictions

The square root function requires its input to be non-negative, so the expression inside must be ≥ 0. The rational function 1/(x+5) is undefined when the denominator is zero, so x ≠ -5. These restrictions must be considered when finding the domain of the composite function.

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