Problem 1 - A Graphical Approach to College Algebra

Question

Fill in each blank with the correct response. Do not use a calculator. The domain and the range of the identity function are both______.

Step-by-Step Solution

Answer

The domain and range are both \( \mathbb{R} \).

Step 1: Identify the Identity Function

The identity function is defined as \( f(x) = x \), meaning that for any input \( x \), the output is exactly \( x \) itself.

Step 2: Determine the Domain

The domain of a function is the set of all possible inputs (\( x \) values) it can accept. Since any real number \( x \) can be input into the identity function \( f(x) = x \), the domain is all real numbers, denoted by \( \mathbb{R} \).

Step 3: Determine the Range

The range of a function is the set of all possible outputs it can produce. Because \( f(x) = x \) outputs exactly what is input (i.e., all real numbers are possible outputs), the range is also all real numbers, denoted by \( \mathbb{R} \).

Step 4: Fill the Blanks

With the domain and range both determined as \( \mathbb{R} \), we can fill in the blanks by stating: 'The domain and the range of the identity function are both \( \mathbb{R} \).'

Key Concepts

Domain

In mathematics, when we talk about the domain of a function, we refer to the set of all possible input values that the function can accept. This is essentially all the values of \( x \) that you can plug into a function without causing any issues, like division by zero or taking the square root of a negative number.
For the identity function, defined as \( f(x) = x \), every real number can be an input. This means that the identity function has no restrictions on its domain.

• Any negative number can be used as an input.
• Zero can also be an input.
• Any positive number is also a valid input.

Therefore, the domain of the identity function is the set of all real numbers, represented mathematically as \( \mathbb{R} \).
The identity function is unique in that it transforms every input directly into the output, making it straightforward to determine its domain.

Range

The range of a function refers to the set of all possible outputs it can generate. It tells us what kind of values the function can produce after we input any of its domain values. Determining the range is often the next step after identifying a function's domain.
In the identity function \( f(x) = x \), each output is exactly the same as the input.
This means that any real number input results in the same real number output.

• If you input \( -5 \), you get \( -5 \) as the output.
• If you input \( 0 \), the output will also be \( 0 \).
• Similarly, if the input is \( 10 \), your output remains \( 10 \).

Thus, the range of the identity function is also the set of all real numbers, expressed as \( \mathbb{R} \).
Since both input and output cover all real numbers, this makes the identity function unique and straightforward.

Real Numbers

The set of real numbers, denoted by \( \mathbb{R} \), includes all the numbers you can place on a number line. This set is crucial to understanding many mathematical functions, including our identity function. Real numbers encompass:

• Natural numbers: 1, 2, 3, and so on.
• Whole numbers: 0, 1, 2, 3, etc.
• Integers: ...,-3, -2, -1, 0, 1, 2, 3,...
• Rational numbers: Numbers that can be expressed as a fraction of two integers, like \( \frac{1}{2} \) or \( -\frac{4}{3} \).
• Irrational numbers: Numbers that cannot be expressed as simple fractions, such as \( \pi \) or \( \sqrt{2} \).

The completeness of real numbers allows for functions such as the identity function to have a domain and range that covers all real numbers. This ensures there are no breaks or gaps in the input-output relationship for the identity function. It’s a broad and inclusive set of numbers that provides a solid foundation for various mathematical operations and concepts.