What is a Range of a Function?

S3-SA5-0196

Grade Level:

Class 10

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Range of a function is the set of all possible output values (y-values) that the function can produce. It tells us 'what answers we can get' from the function when we put in different valid inputs.

Simple Example

Quick Example

Imagine a mobile recharge shop where the amount you pay (input) determines the talk time you get (output). If paying Rs 100 gives 50 minutes, Rs 200 gives 100 minutes, and Rs 300 gives 150 minutes, then the set of all possible talk times (50, 100, 150 minutes) would be the range for these specific inputs.

Worked Example

Step-by-Step

Let's find the Range of the function f(x) = x + 2, for the input values (Domain) {1, 2, 3}.

Step 1: Substitute the first input value, x = 1, into the function.
f(1) = 1 + 2 = 3

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Step 2: Substitute the second input value, x = 2, into the function.
f(2) = 2 + 2 = 4

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Step 3: Substitute the third input value, x = 3, into the function.
f(3) = 3 + 2 = 5

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Step 4: Collect all the output values obtained.
The output values are {3, 4, 5}.

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Step 5: State the Range.
The Range of the function f(x) = x + 2 for the given domain is {3, 4, 5}.

Why It Matters

Understanding the Range helps scientists, engineers, and data analysts predict possible outcomes and limits. For example, in AI/ML, knowing the range of a model's output helps ensure it gives sensible predictions. It's crucial for careers in data science and even designing safe engineering systems.

Common Mistakes

MISTAKE: Confusing Range with Domain. | CORRECTION: Domain is the set of all possible inputs (x-values), while Range is the set of all possible outputs (y-values). Remember: 'D' for 'do' (input), 'R' for 'result' (output).

MISTAKE: Forgetting to list unique values in the Range if an output repeats. | CORRECTION: The Range is a set, so each distinct output value should be listed only once, even if it appears multiple times.

MISTAKE: Not considering all possible inputs when the domain is not explicitly given (e.g., all real numbers). | CORRECTION: If the domain isn't restricted, think about what outputs are possible for *any* valid input. For example, x^2 will always give non-negative results, so its range would be all non-negative numbers.

Practice Questions

Try It Yourself

QUESTION: Find the Range of the function f(x) = 2x for the domain {0, 1, 2}. | ANSWER: {0, 2, 4}

QUESTION: A function g(x) = x^2 takes inputs from the set {-2, -1, 0, 1, 2}. What is its Range? | ANSWER: {0, 1, 4}

QUESTION: For the function h(x) = 10 - x, if the domain is all natural numbers less than 5, what is the Range? (Natural numbers start from 1). | ANSWER: {6, 7, 8, 9}

MCQ

Quick Quiz

Which of the following sets represents the Range of the function f(x) = x + 5, if the domain is {0, 5, 10}?

{0, 5, 10}

{5, 10, 15}

{0, 10, 15}

{5, 15}

The Correct Answer Is:

For x=0, f(0)=0+5=5. For x=5, f(5)=5+5=10. For x=10, f(10)=10+5=15. The set of output values is {5, 10, 15}, which is the Range.

Real World Connection

In the Real World

Think about online food delivery apps like Zomato or Swiggy. When you enter your location (input), the app shows you restaurants that can deliver to you (output). The 'range' here would be the set of all possible delivery zones that a particular restaurant covers. This helps them manage logistics and customer expectations efficiently.

Key Vocabulary

Key Terms

FUNCTION: A rule that assigns each input exactly one output | DOMAIN: The set of all possible input values for a function | OUTPUT: The value produced by a function for a given input | SET: A collection of distinct objects

What's Next

What to Learn Next

Great job understanding the Range! Next, you should explore 'Types of Functions' and 'Graphing Functions'. Knowing the Range will make it much easier to visualize functions and understand their behavior on a graph.