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Chapter 1: Problem 96

Describe the relationship between the graph of a function and the graph of its inverse function.

Short Answer

Expert verified
The relationship between the graph of a function and the graph of its inverse function is shown by the graph of the inverse function being a reflection of the graph of the original function across the line y=x. If the graph of function f crosses the point (a, b), then the graph of the inverse function f^-1 crosses the point (b, a) and vice versa.

Step by step solution

01

Understand the inverse function

The inverse function is a concept in mathematics that 'reverses' the inputs and outputs of the initial function. If the original function turns x into y, the inverse function turns y back into x. This is akin to something like undoing the original function.
02

Reflect across the line y=x

Graphically, the inverse function appears as a reflection of the original function about the line y=x. This means that every point (x, y) on the graph of the original function corresponds to point (y, x) on the graph of the inverse function.
03

Confirm the relationship

Through the observation from steps 1 and 2, we can confirm the relationship between a function and its inverse. If the function f takes x to y, then the inverse function (usually denoted as f^-1) takes y back to x. This is shown graphically as the reflection of the function across the line y=x. If the graph of function f crosses the point (a, b), then the graph of the inverse function f^-1 crosses the point (b, a), and vice versa.

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Most popular questions from this chapter

Prove that a function has an inverse function if and only if it is one-to-one

Use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ f(x)=x^{3}+x-1 $$

A dial-direct long distance call between two cities costs \(\$ 1.04\) for the first 2 minutes and \(\$ 0.36\) for each additional minute or fraction thereof. Use the greatest integer function to write the cost \(C\) of a call in terms of time \(t\) (in minutes). Sketch the graph of this function and discuss its continuity.

Prove or disprove: if \(x\) and \(y\) are real numbers with \(y \geq 0\) and \(y(y+1) \leq(x+1)^{2},\) then \(y(y-1) \leq x^{2}\)

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}+x-1, \quad[0,5], \quad f(c)=11 $$
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