Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7.
$ƒ (x) = √ (x - 1), g (x) = 3 x$

검증된 단계별 안내

Identify the given functions: $f(x) = \sqrt{x - 1}$ and $g(x) = 3x$.

Find the composition $(f \circ g)(x)$, which means $f(g(x))$. Substitute $g(x)$ into $f$: $(f \circ g)(x) = f(3x) = \sqrt{3x - 1}$.

Determine the domain of $(f \circ g)(x)$ by considering the domain restrictions of $f$ after substitution. Since $f(x) = \sqrt{x - 1}$ requires $x - 1 \geq 0$, set the inside of the square root in the composition $3x - 1 \geq 0$.

Solve the inequality $3x - 1 \geq 0$ to find the domain of $(f \circ g)(x)$. This gives $x \geq \frac{1}{3}$.

Conclude that the domain of $(f \circ g)(x)$ is all real numbers $x$ such that $x \geq \frac{1}{3}$.

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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 (f∘g)(x) = f(g(x)). It means you first evaluate g at 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) includes all x-values in the domain of g such that g(x) lies in the domain of f. Identifying these restrictions ensures the composition is valid.

Square Root Function Domain Restrictions

For functions involving square roots, the expression inside the root must be non-negative to produce real outputs. For f(x) = √(x - 1), the domain requires x - 1 ≥ 0, or x ≥ 1. This restriction affects the domain of the composition when substituting g(x) into f.

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7.ƒ(x)=√(x−1),g(x)=3xƒ(x)=√(x-1), g(x)=3x