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Chapter 1: Problem 56

Multiple Choice Which of the following is the inverse of \(f(x)=3 x-2 ? \mathrm{}\)

Short Answer

Expert verified
The inverse of the function \( f(x)=3x-2 \) is \( f^{-1}(x) = (x + 2) / 3 \).

Step by step solution

01

Replace function notation

Firstly, replace \( f(x) \) with \( y \). So, the equation becomes: \( y = 3x - 2 \)
02

Interchange x and y

Next, interchange \( x \) and \( y \). This yields: \( x = 3y - 2 \)
03

Solve the equation for y

Lastly, solve the equation for \( y \), our new \( f(x) \) after interchanging. Add 2 to both sides of the equation to get: \( x + 2 = 3y \). Then divide each side by 3 to get: \( y = (x + 2) / 3 \)
04

Convert back to function notation

After finding the value of \( y \), convert back to function notation. So, the inverse function, \( f^{-1}(x) \), is: \( f^{-1}(x) = (x + 2) / 3 \)

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Key Concepts

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

Function Notation
In mathematics, function notation is a simple yet powerful way to denote functions. It allows us to express relationships between variables clearly.
For example, in the given exercise, the original function is denoted as \( f(x) = 3x - 2 \). Here, \( f \) represents the function, and \( x \) is the input variable. The expression \( 3x - 2 \) defines how the output (or function value) is calculated from the input.
Using function notation is helpful because:
  • It preserves consistency when working with multiple functions.
  • It provides clarity on the variable relationships over various equations.
Changing back and forth between different symbols, like from \( f(x) \) to \( y \) and back, can be essential when solving for inverse functions or manipulating the equations.
Solving Equations
Solving equations is a fundamental skill in mathematics, especially when working with functions.
In the original exercise, after interchanging the variables, we need to solve the new equation "\( x = 3y - 2 \)" to express \( y \) in terms of \( x \).
This involves performing operations to isolate \( y \) on one side of the equation:
  • Add 2 to both sides: \( x + 2 = 3y \)
  • Then, divide both sides by 3 to solve for \( y \): \( y = (x + 2) / 3 \)
Solving equations involves:
  • Understanding inverse operations such as addition with subtraction and multiplication with division.
  • Keeping the equation balanced by performing the same operation on both sides.
Being able to solve these equations is crucial for finding inverse functions, as it enables us to reverse the process of the original function.
Interchanging Variables
Interchanging variables is an essential step in finding an inverse function, indicating the reversal of roles between the dependent and independent variables.
In the exercise provided, after substituting \( f(x) \) with \( y \), we interchange the variables \( x \) and \( y \).
Here's why it's necessary:
  • By swapping \( x \) and \( y \), we switch the roles of input and output. This helps us develop the equation needed to find the inverse.
  • The original function defines how a specific input leads to a specific output. The inverse function uncovers how a particular output can lead to its corresponding input.
While interchanging may seem trivial, it's a critical concept in understanding the essence of inverse functions. This method allows us to find solutions where the relationship between variables originally presented by the function is now inverted.

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Most popular questions from this chapter

Multiple Choice Which of the following is the measure of \(\tan ^{-1}(-\sqrt{3})\) in degrees? \(\begin{array}{llll}{\text { (A) }-60^{\circ}} & {\text { (B) }-30^{\circ}} & {\text { (C) } 30^{\circ}} & {\text { (D) } 60^{\circ}} & {\text { (E) } 120^{\circ}}\end{array}\)

Writing to Learn For a curve to be symmetric about the \(x\) -axis, the point \((x, y)\) must lie on the curve if and only if the point \((x,-y)\) lies on the curve. Explain why a curve that is symmetric about the \(x\) -axis is not the graph of a function, unless the function is \(y=0 .\)

Let \(y_{1}=x^{2}\) and \(y_{2}=2^{x}\) . (a) Graph \(y_{1}\) and \(y_{2}\) in \([-5,5]\) by \([-2,10] .\) How many times do you think the two graphs cross? (b) Compare the corresponding changes in \(y_{1}\) and \(y_{2}\) as \(x\) changes from 1 to \(2,2\) to \(3,\) and so on. How large must \(x\) be for the changes in \(y_{2}\) to overtake the changes in \(y_{1} ?\) (c) Solve for \(x : x^{2}=2^{x}\) . \(\quad\) (d) Solve for \(x : x^{2}<2^{x}\)

In Exercises \(31-36,\) solve the equation in the specified interval. $$\sec x=-3\( \)-\pi\( \)\leq x<\( \)\pi$$

In Exercises \(27-30\) , give the measure of the angle in radians and degrees. Give exact answers whenever possible. $$\sin ^{-1}(0.5)$$
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