What Are the Domain and Range of a Function?

TL;DR

The domain of a function includes all valid input values, while the range consists of all possible outputs. For example, a function charging a minimum fee has a domain of non-negative rational numbers and a range that reflects minimum charges. Different functions can have varied domains and ranges based on their equations.

Transcript

Let's do some example problems dealing with functions and their domains and ranges. Just as a review, a function is just an operator-- let's say this function is f; that tends to be the most typical letter for functions-- that operates on some input, in this case, the input is x, and it produces some output y. Or you could view it as you take some ... Read More

Key Insights

• 😑 Functions have inputs (domain) and outputs (range), and they can be represented by equations or expressions.
• ⚾ The domain of a function can be restricted to specific values based on the given context.
• 🧡 Different functions can have different domains and ranges depending on the nature of the function and its equation.
• 🧡 The range of a function represents all the possible values that can be obtained as outputs.
• 🧡 In some cases, the range may have limitations or restrictions based on the nature of the function and its equation.

Questions & Answers

Q: What is the function definition for Dustin's lawn mowing charges?

Dustin's function, d(hours), is equal to 10 times the number of hours worked (10h), with the hours (h) being non-negative. For example, if he works half an hour, he would charge $5.

Q: What are the domain and range for Maria's tutoring charges?

The domain for Maria's function, m(hours), is non-negative rational numbers. The range is rational numbers greater than or equal to $15, as she has a minimum charge of $15 regardless of the number of hours.

Q: What is the domain and range for the function f(x) = 15x - 12?

The domain for this function is all real numbers, and the range is also all real numbers.

Q: What is the function definition for the given function f(x) = 2x^2 + 5?

The function f(x) is equal to 2 times x squared plus 5 (2x^2 + 5), and it can take any real number as input, with the output being all real numbers greater than or equal to 5.

Q: What is the domain and range for the function f(x) = 1/x?

The domain for this function is all real numbers except zero, as dividing by zero is undefined. The range is also all real numbers except zero, as the function can take any non-zero value as input.

Q: What are the values in the range of the function f(x) = x^2 - 5 when x is restricted to -2, -1, 0, 1, and 2?

When we substitute each value of x (-2, -1, 0, 1, and 2) into the function f(x) = x^2 - 5, we get the range values of -1, -4, -5, -4, and -1, respectively.

Summary & Key Takeaways

• Functions operate on inputs to produce outputs, with the set of input values called the domain and the set of output values called the range.

• Example 1: Dustin charges $10 per hour for mowing lawns, so his function is d(hours) = 10h. The domain is non-negative rational numbers, and the range is non-negative rational numbers greater than or equal to $0.

• Example 2: Maria charges $25 per hour for tutoring math, with a minimum charge of $15. Her function is m(hours) = $15 (if hours < 3/5) or $25h (if hours ≥ 3/5). The domain is non-negative rational numbers, and the range is rational numbers greater than or equal to $15.

• Example 3: The function f(x) = 15x - 12 has a domain of all real numbers and a range of all real numbers.

• Example 4: The function f(x) = 2x^2 + 5 has a domain of all real numbers and a range of all real numbers greater than or equal to 5.

• Example 5: The function f(x) = 1/x has a domain of all real numbers except zero and a range of all real numbers except zero.

• Example 6: Given the domain -2, -1, 0, 1, 2 and the function f(x) = x^2 - 5, the range includes -4, -1, and -5.