More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

G o G

검증된 단계별 안내

Step 1: Understand the notation \( G \circ G \). This represents the composition of the function \( G(x) = \frac{1}{x-2} \) with itself, meaning \( G(G(x)) \).

Step 2: Substitute \( G(x) \) into itself. Start by replacing \( x \) in \( G(x) \) with \( G(x) \). This gives \( G(G(x)) = G\left(\frac{1}{x-2}\right) = \frac{1}{\left(\frac{1}{x-2}\right) - 2} \).

Step 3: Simplify the expression \( \frac{1}{\left(\frac{1}{x-2}\right) - 2} \). To do this, find a common denominator for the terms in the denominator: \( \frac{1}{x-2} - 2 = \frac{1 - 2(x-2)}{x-2} = \frac{1 - 2x + 4}{x-2} = \frac{5 - 2x}{x-2} \).

Step 4: The expression becomes \( G(G(x)) = \frac{x-2}{5-2x} \).

Step 5: Determine the domain of \( G(G(x)) \). The domain of \( G(G(x)) \) is all real numbers except where the denominator is zero. Solve \( 5 - 2x = 0 \) to find the values to exclude from the domain. Also, consider the domain of \( G(x) \) itself, which excludes \( x = 2 \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Composite Functions

A composite function is formed when one function is applied to the result of another function. It is denoted as (f o g)(x) = f(g(x)). Understanding how to combine functions is essential for evaluating composite functions, as it requires substituting the output of one function into another.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When dealing with composite functions, it is crucial to determine the domain of each individual function and how they interact, as the domain of the composite function may be restricted by the domains of the functions involved.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is important in the context of composite functions, as it can affect the overall behavior and domain of the resulting composite function, particularly when combined with other functions that may have restrictions.

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.G o G