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Chapter 4: Problem 40

Find the inverse function of \(f\). $$f(x)=-x$$

Short Answer

Expert verified
The inverse function is \(f^{-1}(x) = -x\).

Step by step solution

01

Understanding the Problem

We are given a function \(f(x) = -x\) and need to find its inverse function \(f^{-1}(x)\). Recall that the inverse function is defined such that if \(y = f(x)\), then \(x = f^{-1}(y)\). This means we need to express \(x\) in terms of \(y\).
02

Replacing f(x) with y

Start by rewriting the function \(f(x) = -x\) in terms of \(y\). Set \(y = -x\). This helps to distinguish between the variables when finding the inverse.
03

Solve for x in terms of y

The next step is to solve the equation \(y = -x\) for \(x\). To do this, multiply both sides by -1, giving us \(x = -y\).
04

Express the Inverse Function

From the previous step, we found that \(x = -y\). Therefore, the inverse function is expressed by swapping the variables back such that \(y = f^{-1}(x) = -x\). Thus, the inverse is \(f^{-1}(x) = -x\).
05

Verify the Result

Verify the inverse function by checking that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). Both checks prove true:1. \(f(f^{-1}(x)) = f(-x) = -(-x) = x\)2. \(f^{-1}(f(x)) = f^{-1}(-x) = -(-x) = x\)Thus, our inverse function \(f^{-1}(x) = -x\) is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

One-to-One Functions
Understanding if a function is one-to-one is crucial in determining whether an inverse function exists. A function is considered one-to-one if each input maps to a unique output. This means that no two different inputs should produce the same output. In simpler terms, a horizontal line should intersect the graph of the function at most once.
  • This property ensures that we can reverse the process uniquely, making the inverse function possible.
  • For example, the function \(f(x) = -x\) is one-to-one because every value of \(x\) produces a different \(f(x)\).
By knowing a function is one-to-one, we can proceed with finding its inverse with confidence, ensuring the inverse will also map its range back to the original domain uniquely.
Function Composition
Function composition involves applying one function to the results of another, forming a linkage. It provides a verification method for calculating inverse functions. If \(f\) and \(g\) are two functions, then their composition is written as \(f(g(x))\).
  • When verifying inverses, we check if \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
  • This ensures that the functions truly undo each other.
In our exercise, using function composition, we confirmed that \(f(-x) = x\) and \(-f(x) = x\), proving that the given inverse function \(f^{-1}(x) = -x\) is indeed correct, as it restores the original input value \(x\) after mapping through \(f\).
Domain and Range
The domain of a function is the complete set of possible input values, while the range is the set of possible output values.
  • Understanding these helps in determining the boundaries and applicability of the inverse function.
  • The original function \(f(x) = -x\) has a domain and range of all real numbers, denoted as \(-\infty , \infty\).
For the inverse function \(f^{-1}(x) = -x\), the domain becomes the range of the original function, and the range becomes the domain, basically swapping the two. This swap makes sense since an inverse function reverses input-output pairs.
Algebraic Manipulation
Algebraic manipulation involves changing the form of a mathematical expression to solve equations or simplify expressions. It is essential in finding inverse functions efficiently.
  • In our exercise, we used simple algebraic steps to find the inverse of the function \(f(x) = -x\).
  • This involved rewriting the function and solving for \(x\) by multiplying both sides by -1, yielding \(x = -y\).
  • The final step swapped variables back, confirming the inverse \(f^{-1}(x) = -x\).
Being comfortable with basic algebraic operations makes finding inverses more straightforward and intuitive, as it often involves simple steps of solving and rearranging equations.

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Most popular questions from this chapter

For manufacturers of computer chips, it is important to consider the fraction \(F\) of chips that will fail after \(t\) years of service. This fraction can sometimes be approximated by the formula \(F=1-e^{-c t},\) where \(c\) is a positive constant. (a) How does the value of \(c\) affect the reliability of a chip? (b) If \(c=0.125,\) after how many years will 35 \(\%\) of the chips have failed?

Sketch the graph of \(f\). $$f(x)=\ln |x|$$

Approximate the function at the value of \(x\) to four decimal places. (a) \(f(x)=2^{3 / 1-x}, \quad x=2.5\) (b) \(g(x)=\left(\frac{2}{25}+x\right)^{-3 x}, \quad x=2.1\) \(h(x)=\frac{3^{-x}+5}{3^{x}-16}, \quad x=\sqrt{2}\)

In a survey of 15 cities ranging in population \(P\) from 300 to \(3,000,000,\) it was found that the average walking speed \(S\) (in \(\mathrm{ft} / \mathrm{sec}\) ) of a pedestrian could be approximated by \(S=0.05+0.86 \log P\) (a) How does the population affect the average walking speed? (b) For what population is the average walking speed \(5 \mathrm{ft} / \mathrm{sec} ?\)

Exer. \(65-66:\) Graph \(f\) on the interval \([0.2,16]\). (a) Estimate the intervals where \(f\) is increasing or is decreasing. (b) Estlmate the maximum and minimum values of \(f\) on \([0.2,16]\) $$f(x)=2 \log 2 x-1.5 x+0.1 x^{2}$$
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