What is the Range of a Function?

S3-SA5-0238

Grade Level:

Class 9

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 that the function can produce. Think of it as all the answers you can get when you put different valid inputs into a function machine. It tells you 'what comes out' of the function.

Simple Example

Quick Example

Imagine a juice shop where the function is 'making juice'. If you only have apples and oranges as fruits (inputs), and the machine only makes apple juice and orange juice (outputs), then the 'Range' of this juice-making function is just {apple juice, orange juice}. You won't get grape juice if you don't put in grapes!

Worked Example

Step-by-Step

Let's find the Range of the function f(x) = x + 2, if the allowed inputs (Domain) are {1, 2, 3, 4}.

STEP 1: Take the first input from the Domain, which is 1.
---STEP 2: Substitute x = 1 into the function: f(1) = 1 + 2 = 3.
---STEP 3: Take the next input, 2. Substitute x = 2: f(2) = 2 + 2 = 4.
---STEP 4: Take the next input, 3. Substitute x = 3: f(3) = 3 + 2 = 5.
---STEP 5: Take the last input, 4. Substitute x = 4: f(4) = 4 + 2 = 6.
---STEP 6: Collect all the output values we found: {3, 4, 5, 6}.
---ANSWER: The Range of the function f(x) = x + 2 for the given Domain is {3, 4, 5, 6}.

Why It Matters

Understanding the Range helps scientists and engineers know the possible outcomes of their experiments or designs. In Data Science, knowing the Range of data helps predict trends, like how much a stock price might change or the possible scores a cricket team can achieve. This skill is crucial for careers in AI, data analysis, and even designing safe bridges!

Common Mistakes

MISTAKE: Confusing Range with Domain. Students sometimes list the input values as the Range. | CORRECTION: Remember, the Range is always about the OUTPUT values (the results you get), while the Domain is about the INPUT values (what you put in).

MISTAKE: Forgetting to list all possible output values or listing duplicates. | CORRECTION: Always calculate the output for EACH valid input in the Domain. If an output value repeats, list it only once in the set for the Range.

MISTAKE: Assuming the Range will always be a continuous set of numbers (e.g., all numbers between 1 and 10) even for discrete inputs. | CORRECTION: If the Domain is a specific set of distinct numbers (like {1, 2, 3}), the Range will also be a specific set of distinct numbers, not necessarily a continuous interval.

Practice Questions

Try It Yourself

QUESTION: A mobile data plan charges Rs. 10 per GB used. If a user can use 1 GB, 2 GB, or 3 GB, what is the Range of total cost? | ANSWER: {Rs. 10, Rs. 20, Rs. 30}

QUESTION: Find the Range of the function g(x) = x * x (or x^2) if the Domain is {-2, -1, 0, 1, 2}. | ANSWER: {0, 1, 4}

QUESTION: A function h(x) gives the number of runs scored by a batsman in an over, where x is the number of balls faced (0 to 6). If the batsman can only score 0, 1, 2, 3, 4, or 6 runs per ball, and faces 6 balls, what is the Range of possible total runs for that over? (Assume no extras). | ANSWER: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} (All whole numbers from 0 to 36, as every combination is possible)

MCQ

Quick Quiz

What is the Range of the function f(x) = x - 5, if the Domain is {6, 7, 8}?

{1, 2, 3}

{6, 7, 8}

{1, 2, 3, 4, 5}

{-1, 0, 1}

The Correct Answer Is:

For x=6, f(6) = 6-5 = 1. For x=7, f(7) = 7-5 = 2. For x=8, f(8) = 8-5 = 3. So the set of all output values is {1, 2, 3}.

Real World Connection

In the Real World

When you use a food delivery app like Swiggy or Zomato, the 'Range' of available food items changes based on your location (the Domain). If you're in Bengaluru, the Range includes Masala Dosa and Filter Coffee, but if you're in Amritsar, it might include Chole Bhature and Lassi. The app's function outputs food options based on your input location.

Key Vocabulary

Key Terms

FUNCTION: A rule that assigns exactly one output for each input. | DOMAIN: The set of all possible input values for a function. | OUTPUT: The result obtained after applying a function to an 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' like linear, quadratic, and polynomial functions. This will help you see how different function rules lead to different kinds of Ranges, making your mathematical journey even more exciting!