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Chapter 11: Problem 51

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. \(f(x)=2 x^{3}-3\)

Short Answer

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

Step by step solution

01

Understand the Function

We are given the function \( f(x) = 2x^3 - 3 \). Our goal is to find its inverse, which we will denote as \( f^{-1}(x) \).
02

Replace Function Notation with Output Variable

Replace \( f(x) \) with \( y \), so the function becomes \( y = 2x^3 - 3 \).
03

Swap Variables

To find the inverse, swap \( x \) and \( y \). The equation becomes \( x = 2y^3 - 3 \).
04

Solve for the Original Input Variable

Solve the equation for \( y \):1. Add 3 to both sides: \( x + 3 = 2y^3 \).2. Divide both sides by 2: \( \frac{x + 3}{2} = y^3 \).3. Take the cube root of both sides: \( y = \sqrt[3]{\frac{x + 3}{2}} \).
05

Write the Inverse Function

The inverse function is therefore \( f^{-1}(x) = \sqrt[3]{\frac{x + 3}{2}} \).

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Key Concepts

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

One-to-One Functions
A one-to-one function is a special type of function where each input corresponds to a unique output. This means that no two different inputs will produce the same output. One way to check if a function is one-to-one is by using the horizontal line test. For this test, you draw horizontal lines across the graph of the function.
  • If any horizontal line intersects the graph more than once, the function is not one-to-one.
  • If no horizontal line intersects the graph more than once, the function is one-to-one.
An example is the function given in the exercise, \( f(x) = 2x^3 - 3 \). Each input of \( x \) produces a distinct output, confirming it's one-to-one. This property is crucial because only one-to-one functions have inverses that are also functions.
Function Notation
Function notation is a concise way of expressing the idea of a function. It uses symbols to convey the action of a function on an input to produce an output. In the notation \( f(x) \), \( f \) is the name of the function, and \( x \) is the input variable. The output of the function is usually expressed as a mathematical rule applied to \( x \).
In the context of finding an inverse, we first express \( f \) in terms of \( y \) (the output). For example, \( y = 2x^3 - 3 \). This expression helps when swapping \( x \) and \( y \) to find the inverse.
  • Remember: Inverse functions are denoted using \( f^{-1}(x) \).
  • Function notation aids in clearly distinguishing between a function and its inverse.
Cube Root
A cube root is the number that, when multiplied by itself three times, gives the original number. The cube root symbol \( \sqrt[3]{} \) is used to denote this operation.
For example, the cube root of 8 is 2, since \( 2 \times 2 \times 2 = 8 \). Calculating the cube root is essential when solving for an inverse function, especially if the function involves a cube, as demonstrated in the function \( x = 2y^3 - 3 \).
The step requires taking the cube root of both sides to isolate \( y \), resulting in \( y = \sqrt[3]{\frac{x + 3}{2}} \). This move is a crucial part of solving and manipulating equations involving cubes.
Solving Equations
Solving equations involves manipulating expressions to find the value of a variable. It often includes steps like isolating variables, adding or subtracting terms, and applying operations like division or taking roots.
In the exercise, we solve for \( y \) after swapping \( x \) and \( y \) in the equation \( x = 2y^3 - 3 \). This requires:
  • Adding 3 to both sides to simplify: \( x + 3 = 2y^3 \).
  • Dividing both sides by 2 to continue isolating \( y \): \( \frac{x + 3}{2} = y^3 \).
  • Taking the cube root of both sides to find \( y \), resulting in \( y = \sqrt[3]{\frac{x + 3}{2}} \).
Mastering equation solving is key to handling inverse functions and many other mathematical problems.

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Most popular questions from this chapter

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ e^{3 x}=9 $$

a. \(5^{9 x-1}=125\) b. \(5^{9 x-1}=124\)

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ 8^{x^{2}}=11 $$

Rule of Seventy. A rule of thumb for finding how long it takes an investment to double is called the rule of seventy. To apply the rule, divide 70 by the interest rate written as a percent. At \(5 \%\), an investment takes \(\frac{70}{5}=14\) years to double. At \(7 \%,\) it takes \(\frac{70}{7}=10\) years. Explain why this formula works.

Use a calculator to evaluate each expression, if possible. Express all answers to four decimal places. See Using Your Calculator: Evaluating Base-e (Natural) Logarithms. $$\ln 2.7$$
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