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Chapter 5: Problem 60

For each function: a) Determine whether it is one-to-one. b) If the function is one-to-one, find a formula for the inverse. $$f(x)=-\frac{3}{x}$$

Short Answer

Expert verified
The function is one-to-one. The inverse is \( f^{-1}(x) = -\frac{3}{x} \).

Step by step solution

01

- Define the function

Given the function is \( f(x) = -\frac{3}{x} \).
02

- Examine if the function is one-to-one

A function is one-to-one if every value of y in the range corresponds to exactly one value of x in the domain. To determine if \( f(x) \) is one-to-one, we check if each pair of different inputs produces different outputs. Suppose \( f(x_1) = f(x_2) \). This gives us: \[ -\frac{3}{x_1} = -\frac{3}{x_2} \] Simplifying, we remove the negative and obtain: \[ \frac{3}{x_1} = \frac{3}{x_2} \] Cross-multiplying results in: \[ 3x_2 = 3x_1 \] Dividing both sides by 3, we get: \[ x_1 = x_2 \] Thus, the function is one-to-one.
03

- Find the inverse function

Since \( f(x) \) is one-to-one, we can find its inverse. To find the inverse, start by replacing \( f(x) \) with y: \[ y = -\frac{3}{x} \] Our goal is to solve for x in terms of y. First, multiply both sides by x: \[ xy = -3 \] Next, solve for x: \[ x = -\frac{3}{y} \] Finally, replace y with \( f^{-1}(x) \): \[ f^{-1}(x) = -\frac{3}{x} \]

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Key Concepts

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

inverse function
In mathematics, an inverse function is a function that reverses the operation of the original function. If you have a function \( f(x) \), its inverse is denoted as \( f^{-1}(x) \). When you apply both a function and its inverse, you'll end up back where you started, which means \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

For the function \( f(x) = -\frac{3}{x} \), to find its inverse, we need to swap the dependent and independent variables and then solve for the new dependent variable. It involves:
  • Rewriting the function as \( y = -\frac{3}{x} \)
  • Swapping y and x so it becomes \( x = -\frac{3}{y} \)
  • Solving for y, resulting in \( y = -\frac{3}{x} \)
  • Thus, the inverse function is \( f^{-1}(x) = -\frac{3}{x} \).


Notice that every function does not necessarily have an inverse. It must be one-to-one, which ensures that each output is produced by a unique input.
function definition
A function is a relation between two sets of numbers or values, where each input (or x-value) from the first set has exactly one output (or y-value) in the second set. For example, the function \( f(x) = -\frac{3}{x} \) describes how each input x is mapped to an output y.

There are some key points for defining functions:
  • The domain is the set of all possible inputs (x-values) that the function can accept.
  • The range is the set of all possible outputs (y-values) that the function can produce.
  • A function is one-to-one if every output is connected to one unique input. Determining whether \( f(x) = -\frac{3}{x} \) is one-to-one involves ensuring no two different inputs give the same output.


By defining functions properly, we can analyze and manipulate them to find important features such as their inverses.
cross-multiplication
Cross-multiplication is a method used to solve equations involving fractions. This technique is extremely helpful when comparing two ratios or when trying to isolate a variable. For example, for the function \( f(x) = -\frac{3}{x} \), determining if it is one-to-one involves comparing two fractions.

Here's how it's done:
  • Start with \( \frac{3}{x_1} = \frac{3}{x_2} \).
  • Cross-multiply by multiplying the numerator of one fraction by the denominator of the other: \( 3x_2 = 3x_1 \).
  • This simplifies to \( x_1 = x_2 \) after dividing both sides by 3.


Cross-multiplication shows that if \( f(x_1) = f(x_2) \), then \( x_1 \) must equal \( x_2 \), confirming that the function is one-to-one.
solving for x
Solving for x means isolating the variable x on one side of an equation. This is a fundamental skill in algebra, necessary for finding inverses or understanding how input values relate to output values.

Let's solve for x in the function \( f(x) = -\frac{3}{x} \) when determining the inverse:
  • Start with \( y = -\frac{3}{x} \).
  • To eliminate the fraction, multiply both sides of the equation by x: \( xy = -3 \).
  • Next, solve for x by dividing both sides by y: \( x = -\frac{3}{y} \).


Finally, when finding the inverse, replace y with \( f^{-1}(x) \) to get \( f^{-1}(x) = -\frac{3}{x} \). This showcases how solving for x can reveal the inverse relationship between variables in a function.

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