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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 27
Chapter 1, Problem 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

Verified step by step guidance
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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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