What is the Range of a Function? | Simple Explanation for Class 10

S6-SA1-0535

What is the Range of a Function (basic introduction)?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

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 values the function 'covers' or 'reaches' when we put in different input values.

Simple Example

Quick Example

Imagine you have a machine that doubles any number you put into it. If you put in numbers like 1, 2, 3, 4, you get out 2, 4, 6, 8. The set {2, 4, 6, 8} is the Range for these specific inputs. If you could put in *any* positive number, the machine would give out *any* positive even number.

Worked Example

Step-by-Step

Let's find the Range for the function f(x) = x + 5, if the allowed input values (Domain) are {1, 2, 3}.

Step 1: Take the first input value, x = 1.
---Step 2: Substitute x = 1 into the function: f(1) = 1 + 5 = 6.
---Step 3: Take the second input value, x = 2.
---Step 4: Substitute x = 2 into the function: f(2) = 2 + 5 = 7.
---Step 5: Take the third input value, x = 3.
---Step 6: Substitute x = 3 into the function: f(3) = 3 + 5 = 8.
---Step 7: Collect all the output values we got.
---The Range of the function for these inputs is {6, 7, 8}.

Why It Matters

Understanding the Range helps scientists and engineers predict outcomes. In AI/ML, it helps define the possible results a model can give. Doctors use it to understand the range of possible drug effects, and physicists use it to determine the limits of physical phenomena.

Common Mistakes

MISTAKE: Confusing Range with Domain (input values). | CORRECTION: Range is about the OUTPUT values (y-values), while Domain is about the INPUT values (x-values).

MISTAKE: Only considering a few input values and assuming that's the full Range. | CORRECTION: For functions with an infinite Domain (like all real numbers), you need to think about the general behaviour of the function to find its true Range, not just test a few points.

MISTAKE: Not considering restrictions on the function (e.g., division by zero, square root of negative numbers). | CORRECTION: Always check if there are any values for which the function is undefined, as these affect both the Domain and the Range.

Practice Questions

Try It Yourself

QUESTION: What is the Range of the function f(x) = 2x, if the Domain is {10, 20, 30}? | ANSWER: {20, 40, 60}

QUESTION: A function gives your exam score (y) based on how many hours you study (x), as y = 10x. If you can study between 0 and 5 hours, what is the Range of your possible scores? | ANSWER: {0, 10, 20, 30, 40, 50} (assuming scores are multiples of 10 for each hour studied)

QUESTION: For the function f(x) = x^2, if the Domain is {-2, -1, 0, 1, 2}, find the Range. | ANSWER: {0, 1, 4}

MCQ

Quick Quiz

If a function is defined as f(x) = x - 3 and its Domain is {5, 6, 7}, what is its Range?

{2, 3, 4}

{5, 6, 7}

{-2, -1, 0}

{8, 9, 10}

The Correct Answer Is:

For x=5, f(5)=5-3=2. For x=6, f(6)=6-3=3. For x=7, f(7)=7-3=4. So the set of all output values is {2, 3, 4}.

Real World Connection

In the Real World

Think about an app like Swiggy or Zomato. When you search for restaurants, the 'range' of delivery times shown (e.g., '20-45 minutes') is essentially the Range of a function that calculates delivery time based on factors like distance and traffic. Or, when ISRO launches a satellite, engineers calculate the range of altitudes it can achieve.

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 value a function produces for a given input | SET: A collection of distinct objects

What's Next

What to Learn Next

Now that you understand the Range, you can explore different types of functions like linear, quadratic, and polynomial functions. Understanding their Ranges will help you solve more complex problems and graph them accurately.