Problem 58 - College Algebra with Corequisite Support

Question

Create a function in which the range is all nonnegative real numbers.

Step-by-Step Solution

Answer

Use the function \( f(x) = |x| \); it outputs only nonnegative numbers.

Step 1: Understand the Problem

Our task is to construct a function whose range, or the set of all possible output values, includes only nonnegative real numbers. This means the function's output should never be negative.

Step 2: Familiarize with Function Types

Common functions that fit the criteria, where the range is nonnegative, include the square function, absolute value function, and the exponential function. These functions are known to yield only nonnegative outputs regardless of the input.

Step 3: Choose a Function

Select a function that is simple and commonly known to have a nonnegative range. A simple choice is the absolute value function, denoted as \( f(x) = |x| \). This function always returns the nonnegative value of the input.

Step 4: Verify the Range

Check if the chosen function, \( f(x) = |x| \), really yields a nonnegative range. By definition, the absolute value of any real number is either positive or zero, confirming that the range includes all nonnegative real numbers.

Step 5: Write the Function

The function is defined as \( f(x) = |x| \). This confirms that for any real number input \( x \), the output is \( |x| \), which is always nonnegative.

Step 6: Validate with Examples

For \( x = 3 \), the output \( f(3) = |3| = 3 \). For \( x = -5 \), the output \( f(-5) = |-5| = 5 \). In both examples, outputs are nonnegative, validating our function choice.

Key Concepts

Absolute Value Function

The absolute value function is a core concept in mathematics. It is defined for any real number input and is written as \( f(x) = |x| \). This function takes each input, whether negative or positive, and outputs its nonnegative magnitude.
Here’s how it works:

• If the input \( x \) is positive, the output is simply \( x \) itself.
• If the input \( x \) is negative, the output is \(-x\), making the value positive.
• If the input \( x \) is zero, the output is also zero.

The output of an absolute value function can never be negative, as it is designed to give the distance of a number from zero on a number line. Thus, it is perfect for constructing functions where the range should include only nonnegative real numbers.

Function Types

Functions in mathematics come in various forms, and they determine the relationship between inputs and outputs. Some functions naturally have nonnegative outputs across their entire domain, making them ideal for scenarios where only nonnegative real numbers are desired.
A few common function types with this characteristic include:

• **Absolute Value Function**: As discussed, \( f(x) = |x| \) ensures all outputs are nonnegative.
• **Square Function**: Denoted by \( f(x) = x^2 \), this function also returns only positive values (or zero), as the square of any real number is nonnegative.
• **Exponential Function**: Functions like \( f(x) = a^x \) (where \( a > 0 \)) never produce negative outputs.

Choosing the right function type depends on the context of the problem you are solving, and each of these has distinct characteristics helpful in different scenarios.

Nonnegative Real Numbers

Nonnegative real numbers are a crucial set of numbers in mathematics. This set includes all real numbers that are either positive or zero. It is denoted by \([0, \, \infty)\). Understanding nonnegative real numbers helps in various mathematical contexts, especially when constructing functions where negative outputs are not allowed. Here are a few key features of nonnegative real numbers:

• **Includes Zero**: Unlike positive numbers, nonnegative numbers include zero, making them broader.
• **Used in Functions**: Functions like absolute value, square, and exponential (with specific bases) naturally restrict their range to nonnegative real numbers.
• **Utility**: In real-life applications, nonnegative numbers are often used to measure quantities such as distance, time, and probability, where negative values do not make sense.

When defining functions with ranges of nonnegative real numbers, it is essential to choose operations and function types that inherently yield such outputs.