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Chapter 1: Problem 106

Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

Short Answer

Expert verified
Given function \(f\) is odd and one-to-one. To prove \(f^{-1}\) is odd, we derived the relation \(f^{-1}(f(-y)) = -y\) by replacing \(x\) with \(f(y)\) in the relation \(f^{-1}(-x) = -f^{-1}(x)\) and using the definition of an odd function.

Step by step solution

01

Defining the one-to-one odd function

Let's consider a function \(f\) that is a one-to-one and odd. This implies that for all \(x\), \(f(-x) = -f(x)\). This is the definition of an odd function.
02

Proving the inverse is odd

Assume that \(f^{-1}(x)\) represents the inverse of the function \(f(x)\). We need to prove that \(f^{-1}(-x) = -f^{-1}(x)\) to show that the inverse function is also odd. To do this, let's replace \(x\) with \(f(y)\) in the relation \(f^{-1}(-x) = -f^{-1}(x)\), this gives us \(f^{-1}(-f(y)) = -f^{-1}(f(y))\). Now, using the oddness property of function \(f\), this simplifies to \(f^{-1}(f(-y)) = -y\). But by definition of the inverse function, \(f^{-1}(f(-y)) = -y )\), which concludes the proof.

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Most popular questions from this chapter

Use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. An overhead garage door has two springs, one on each side of the door (see figure). A force of 15 pounds is required to stretch each spring 1 foot. Because of a pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the combined lifting force applied to the door by the springs when the door is closed.

Find a mathematical model for the verbal statement. \(y\) varies inversely as the square of \(x\)

Determine whether the variation model is of the form \(y=k x\) or \(y=k / x,\) and find \(k .\) Then write \(a\) model that relates \(y\) and \(x\). $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & -3.5 & -7 & -10.5 & -14 & -17.5 \\ \hline \end{array} $$

Use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(f^{-1} \circ f^{-1}\right)(6) $$

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{x+1}{x-2} $$
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