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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 1 - Functions문제 27
1장, 문제 27

Find the inverse function (on the given interval, if specified) and graph both ff and f−1f^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.
f(x)=8−4xf\(\left\)(x\(\right\))=8-4x

검증된 단계별 안내
1
To find the inverse of the function \( f(x) = 8 - 4x \), start by replacing \( f(x) \) with \( y \), so we have \( y = 8 - 4x \).
Swap \( x \) and \( y \) to find the inverse function. This gives us \( x = 8 - 4y \).
Solve for \( y \) in terms of \( x \). Start by isolating \( y \) on one side: \( 4y = 8 - x \).
Divide both sides by 4 to solve for \( y \): \( y = \frac{8 - x}{4} \). This is the inverse function, \( f^{-1}(x) = \frac{8 - x}{4} \).
To graph both \( f(x) = 8 - 4x \) and \( f^{-1}(x) = \frac{8 - x}{4} \), plot them on the same set of axes. Check for symmetry about the line \( y = x \), which is a characteristic of inverse functions.

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주요 개념

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

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y as input and returns x. For a function to have an inverse, it must be one-to-one, meaning that each output is produced by exactly one input. This property ensures that the inverse function is well-defined.
추천 영상:
4:49
Inverse Cosine

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between the input (x-values) and output (y-values). The graph of a function can reveal important characteristics such as intercepts, slopes, and asymptotic behavior. When graphing an inverse function, it is crucial to reflect the original function across the line y = x, which helps to illustrate the symmetry between a function and its inverse.
추천 영상:
가이드 코스
5:53
Graph of Sine and Cosine Function

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. For functions and their inverses, this symmetry is observed about the line y = x. This means that if a point (a, b) lies on the graph of f, then the point (b, a) will lie on the graph of f⁻¹. Recognizing this symmetry is essential for verifying the correctness of the inverse function and understanding the relationship between the two graphs.
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