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Chapter 4: Problem 19

Decide whether each function as graphed or defined is one-to-one. $$y=\sqrt[3]{x+1}-3$$

Short Answer

Expert verified
The function \( y = \frac{x+1}{3} - 3 \) is one-to-one.

Step by step solution

01

- Define One-to-One Function

A function is one-to-one if each input corresponds to exactly one unique output, and each output corresponds to exactly one unique input.
02

- Determine Injectivity

To prove the function is one-to-one, determine if the function is injective. A function is injective if for any two different inputs in the domain, their corresponding outputs are different.
03

- Define the Function

Apply the given function: \[ y = \frac{x+1}{3} - 3 \]
04

- Evaluate the Function

Compare the outputs for different inputs to see if they result in unique values.
05

- Solve Algebraically

Assume that \[ f(a) = f(b) \] Then: \[ \frac{a+1}{3} - 3 = \frac{b+1}{3} - 3 \] By isolating we get: \[ \frac{a+1}{3} = \frac{b+1}{3} \] multiplying both sides by 3; \[a+1 = b+1\] This simplifies further to \[ a = b. \] This shows each input has a unique output.
06

- Conclude

Since the only solution to \[ f(a) = f(b) \] implies \[ a = b \], the function is injective, and thus, the function \[ y = \frac{x+1}{3} - 3 \] is one-to-one.

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Key Concepts

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

Function Definition
A function is a mathematical concept where each input corresponds to one unique output. Think of it like a machine where you input a number, and it gives out another number based on a specific rule. For example, if we have a function defined by \[y = x^2 \], when you input 2, the output is 4.

Important characteristics of a function include:
  • The input is also called the domain.
  • The output is also called the range.
  • Each input value should produce exactly one output value.
Understanding these basic properties helps us dive deeper into specific types of functions, like one-to-one functions.
Injective Function
An injective function, also known as a one-to-one function, is a special type of function. It ensures that each unique input maps to a unique output.

Let鈥檚 break down what makes a function injective:
  • If \[f(a) = f(b) \], then it must be that \[a = b.\]
  • This means no two different inputs \[a\] and \[b\] in the domain will produce the same output.

For example, if we have a function \[f(x) = 2x \], and \[f(2) = 4 \] and \[f(3) = 6 \], notice that different values of \[x\] provide different outputs. Hence, this function is one-to-one.
Algebraic Proof
To prove that a function is one-to-one, we often use an algebraic approach. Let鈥檚 use our original function \[y = \frac{x+1}{3} - 3 \].
Suppose \[f(a) = f(b) \]. We need to show this implies \[a = b \]. Here鈥檚 the step-by-step algebraic proof:
  • Assume \[ \frac{a+1}{3} - 3 = \frac{b+1}{3} - 3 \].
  • First, add 3 to both sides to simplify: \[ \frac{a+1}{3} = \frac{b+1}{3} \].
  • Next, multiply both sides by 3: \[a+1 = b+1 \].
  • Finally, subtract 1 from both sides, giving us \[a = b \].

Since we have shown that \[f(a) = f(b) \] implies \[a = b \], the function \[y = \frac{x+1}{3} - 3 \] is algebraically proven to be injective, or one-to-one.

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Most popular questions from this chapter

Answer each of the following. Suppose \(f(r)\) is the volume (in cubic inches) of a sphere of radius \(r\) inches. What does \(f^{-1}(5)\) represent?

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=\frac{1}{x-3}, \quad x \neq 3$$

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=-4 x+3$$

Find \(f^{-1}(x),\) and give the domain and range. $$f(x)=e^{x+1}-4$$

For individual or collaborative investigation (Exercises \(117-122\) ) Assume \(f(x)=a^{x}\), where \(a>1 .\) Work these exercises in order. If \(f\) has an inverse function \(f^{-1},\) sketch \(f\) and \(f^{-1}\) on the same set of axes.
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