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Chapter 4: Problem 21

Explain why a constant function, such as \(f(x)=3,\) defined over the set of real numbers, cannot be one-to-one.

Short Answer

Expert verified
The function assigns the same output to different inputs, violating the one-to-one requirement.

Step by step solution

01

Identify the definition of a constant function

A constant function is defined as a function that assigns the same output value for every input value. For the function given, we have: \[ f(x) = 3 \]
02

Understand the definition of a one-to-one function

A function is one-to-one (injective) if different inputs always produce different outputs. Mathematically, this means for any two inputs, say \(x_1\) and \(x_2\), if \(x_1 eq x_2\), then \(f(x_1) eq f(x_2)\).
03

Compare the definitions

The given constant function \(f(x) = 3\) assigns the same output (which is 3) to any input \(x\). This implies that no matter what different input values \(x_1\) and \(x_2\) are chosen, \(f(x_1) = f(x_2) = 3\).
04

Determine why the constant function fails the one-to-one criteria

Since the constant function assigns the same output (3) to any different inputs \(x_1\) and \(x_2\), it does not satisfy the condition for a one-to-one function. For any \(x_1\) and \(x_2\) where \(x_1 eq x_2\), we have \(f(x_1) = f(x_2) = 3\), which violates the one-to-one requirement.
05

Conclude that the constant function is not one-to-one

Because the constant function does not produce distinct outputs for distinct inputs, it cannot be considered a one-to-one function.

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Key Concepts

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

one-to-one function
A one-to-one function, also known as an injective function, is a type of function where different input values always correspond to different output values. Think of it this way: if you have two different inputs, say \( x_1 \) and \( x_2 \), then the outputs \( f(x_1) \) and \( f(x_2) \) must also be different. Mathematically, this is expressed as:
  • If \( x_1 eq x_2 \), then \( f(x_1) eq f(x_2) \).

This definition is crucial in understanding why a constant function cannot be one-to-one. If you take any two different inputs and the outputs are the same, the function fails to be one-to-one.
injective function
An injective function is simply another term for a one-to-one function. Both terms mean the exact same thing. When we say a function is injective, we are saying that each input maps to a unique output. There are no two different inputs that will give the same output.To better understand, let's look at an example. Suppose we have a function \( f(x) = 2x \). If we take any two different inputs, say \( x = 1 \) and \( x = 2 \), the outputs will be:
  • For \( x = 1 \), \( f(1) = 2 \times 1 = 2 \).
  • For \( x = 2 \), \( f(2) = 2 \times 2 = 4 \).

Since \( f(1) eq f(2) \), this function is injective. The concept of injectivity is essential in many areas of mathematics, such as linear algebra and analysis.
function definition
Understanding the definition of a function is fundamental. A function is a relation between a set of inputs and a set of possible outputs. Each input is related to exactly one output. This can be written as \( f: X \to Y \) where \( X \) is the set of inputs (also called the domain) and \( Y \) is the set of possible outputs (also called the codomain).For example, consider the function \( f(x) = x^2 \) defined over the set of real numbers. Here, \( X \) is the set of all real numbers, and \( Y \) is the set of all non-negative real numbers. This means no matter what real number you plug into the function, you will always get a non-negative real number as the output.Functions are a foundational part of mathematics, and understanding their properties, such as being one-to-one or injective, helps in deeply grasping their behaviors and applications.

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Most popular questions from this chapter

Growth of Bacteria The growth of bacteria makes it necessary to time-date some food products so that they will be sold and consumed before the bacteria count is too high. Suppose for a certain product the number of bacteria present is given by $$ f(t)=500 e^{0.1 t} $$ where \(t\) is time in days and the value of \(f(t)\) is in millions. Find the number of bacteria present at each time. (a) 2 days (b) 4 days (c) 1 week

Use another type of logistic function. Heart Disease As age increases, so does the likelihood of coronary heart disease (CHD). The fraction of people \(x\) years old with some CHD is modeled by $$ f(x)=\frac{0.9}{1+271 e^{-0.122 x}} $$ (Source: Hosmer, D., and S. Lemeshow, Applied Logistic Regression, John Wiley and Sons.) (a) Evaluate \(f(25)\) and \(f(65) .\) Interpret the results. (b) At what age does this likelihood equal \(50 \% ?\)

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. $$f(x)=\frac{x+1}{x-3}, \quad x \neq 3$$

Deer Population The exponential growth of the deer population in Massachusetts can be calculated using the model $$ f(x)=50,000(1+0.06)^{x} $$ where \(50,000\) is the initial deer population and 0.06 is the rate of growth. \(f(x)\) is the total population after \(x\) years have passed. (a) Predict the total population after 4 yr. (b) If the initial population was \(30,000\) and the growth rate was \(0.12,\) approximately how many deer would be present after 3 yr? (c) How many additional deer can we expect in 5 yr if the initial population is \(45,000\) and the current growth rate is \(0.08 ?\) (IMAGE CANT COPY)

Use the definition of inverses to determine whether \(f\) and \(g\) are inverses. $$f(x)=\frac{-1}{x+1}, \quad g(x)=\frac{1-x}{x}$$
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