Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=3x+8 and g(x) = (x-8)/3

검증된 단계별 안내

First, find the composition \( f(g(x)) \) by substituting \( g(x) \) into \( f(x) \). This means replacing every \( x \) in \( f(x) = 3x + 8 \) with \( g(x) = \frac{x - 8}{3} \). So, write \( f(g(x)) = 3 \left( \frac{x - 8}{3} \right) + 8 \).

Next, simplify the expression for \( f(g(x)) \) by performing the multiplication and addition inside the function. Multiply 3 by \( \frac{x - 8}{3} \) and then add 8.

Then, find the composition \( g(f(x)) \) by substituting \( f(x) \) into \( g(x) \). Replace every \( x \) in \( g(x) = \frac{x - 8}{3} \) with \( f(x) = 3x + 8 \). So, write \( g(f(x)) = \frac{(3x + 8) - 8}{3} \).

Simplify the expression for \( g(f(x)) \) by performing the subtraction in the numerator and then dividing by 3.

Finally, determine whether \( f \) and \( g \) are inverses by checking if both compositions \( f(g(x)) \) and \( g(f(x)) \) simplify to \( x \). If both equal \( x \), then \( f \) and \( g \) are inverse functions.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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)) or g(f(x)). It requires substituting the entire expression of one function into the variable of the other, allowing us to combine functions and analyze their combined effect.

Inverse Functions

Inverse functions reverse the effect of each other, meaning f(g(x)) = x and g(f(x)) = x for all x in the domain. To verify if two functions are inverses, we check if their compositions yield the identity function, which returns the input unchanged.

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions, substituting variables, and solving equations. It is essential for correctly performing function composition and verifying inverse relationships by simplifying the composed functions to check if they equal x.