Physics and Mathematics
Range of Function
1. Concept Overview
When we study a function, we always ask two basic questions:
- Which values of x can we put? → Domain
- Which values does the function finally produce? → Range
So, the range of a function tells us:
“What are all the possible outputs of the function?”
2. Definition of Range
The range of a function [f(x)] is the set of all values of [y] such that:
[y = f(x)] for some [x] belonging to the domain.
In simple words:
Range = all possible values of [f(x)]
3. Difference Between Domain and Range
| Domain | Range |
|---|---|
| Input values | Output values |
| x-values | y-values |
| Decided before applying function | Obtained after applying function |
4. Simple Examples
- If [f(x) = x²]
- Domain: all real numbers
- Range: [{[}0, ∞) ]
- If [f(x) = √x]
- Domain: [x ≥ 0]
- Range: [{[}0, ∞) ]
- If [f(x) = 3]
- Range: [{3}]
5. Important Observations
- Range depends on the nature of the function
- Range is always a subset of real numbers
- Every value in the range must be actually attainable
- Range may be finite or infinite
6. How Range Is Affected
- Square functions give non-negative outputs
- Modulus functions give non-negative outputs
- Constant functions give only one value
- Square root functions give restricted outputs
7. Examples with Solutions
Example 1. Find the range of [f(x) = x²].
Solution:
For any real x, the value of x² is always non-negative.
The minimum value occurs at [x = 0].
Range:
[ [0, ∞) ]
Example 2. Find the range of [f(x) = √x].
Solution:
Square root values are never negative.
Also, √x increases without bound as x increases.
Range:
[ [0, ∞) ]
Example 3. Find the range of [f(x) = 3].
Solution:
This is a constant function.
Its output is always 3.
Range:
[{3}]
Example 4. Find the range of [f(x) = |x|].
Solution:
Modulus gives the non-negative value of x.
Minimum value is 0.
Range:
[ [0, ∞) ]
Example 5. Find the range of [f(x) = x − 5].
Solution:
This is a linear function.
As x takes all real values, f(x) also takes all real values.
Range:
[ (−∞, ∞) ]
8. Conceptual Questions with Solutions
1. What is meant by the range of a function?
It is the set of all possible output values of the function.
2. Is the range decided before or after applying the function?
It is decided after applying the function to the domain.
3. Can the range contain values not produced by the function?
No. Every value in the range must be actually obtained.
4. What is the range of f(x) = x²?
Since squares are always non-negative, the range is [0, ∞).
5. What is the range of a constant function?
It contains only one value.
6. Can the range be empty?
No. A function must have at least one output.
7. Can two different x values give the same range value?
Yes, in a many–one function.
8. Is range always equal to codomain?
No. Range is only a subset of the codomain.
9. What decides the range of √x?
The fact that square roots are never negative.
10. Does domain restriction affect range?
Yes. Restricting the domain can change the range.
11. Can the range be infinite?
Yes, for functions like f(x) = x.
12. Is range always an interval?
No. It can be a discrete set as well.
13. How is range seen graphically?
By observing the y-values of the graph.
14. Is 0 always included in the range?
No. It depends on the function.
15. Why is range important?
It helps in solving equations, inequalities, and inverses.
9. FAQ / Common Misconceptions
1. Range and domain are the same.
False. Domain is input, range is output.
2. Range is decided arbitrarily.
False. It is strictly decided by the function rule.
3. Range always equals codomain.
False. Range is usually a subset of codomain.
4. Negative values are always in the range.
False. Many functions never give negative outputs.
5. If x is large, range is also large.
Not always. It depends on the type of function.
6. Constant functions have no range.
False. They have a single-valued range.
7. Range can include values not achieved.
False. Only achieved values are included.
8. Range cannot be a single number.
False. Constant functions prove otherwise.
9. Range is always continuous.
False. It can be discrete.
10. Range is not important in exams.
False. It is highly tested in Class 12.
10. Practice Questions with Step-by-Step Solutions
Question 1. Find the range of [f(x) = x + 2].
Step-by-Step Solution:
The function is linear.
Linear functions take all real values.
Conclusion:
Range = [ (−∞, ∞) ]
Question 2. Find the range of [f(x) = x² + 1].
Step-by-Step Solution:
Since x² ≥ 0 for all x,
[x² + 1 ≥ 1].
Conclusion:
Range = [{[}1, ∞) ]
Question 3. Find the range of [f(x) = √(x − 2)].
Step-by-Step Solution:
Square root values are always non-negative.
Minimum occurs when [x − 2 = 0].
Conclusion:
Range = [{[}0, ∞) ]
Question 4. Find the range of [f(x) = |x − 3|].
Step-by-Step Solution:
Modulus always gives non-negative values.
Minimum value is 0.
Conclusion:
Range = [{[}0, ∞) ]
Question 5. Find the range of [f(x) = 5 − x²].
Step-by-Step Solution:
Since x² ≥ 0,
[5 − x² ≤ 5].
Conclusion:
Range = [ (−∞, 5{]} ]
Question 6. Find the range of [f(x) = 2|x|].
Step-by-Step Solution:
|x| ≥ 0 for all x.
So, [2|x| ≥ 0].
Conclusion:
Range = [ {[}0, ∞) ]
Question 7. Find the range of [f(x) = √(x²)].
Step-by-Step Solution:
√(x²) = |x|.
Modulus is always non-negative.
Conclusion:
Range = [ {[}0, ∞) ]
Question 8. Find the range of [f(x) = x² − 4].
Step-by-Step Solution:
Since x² ≥ 0,
[x² − 4 ≥ −4].
Conclusion:
Range = [ {[}−4, ∞) ]
Question 9. Find the range of [f(x) = −|x|].
Step-by-Step Solution:
|x| ≥ 0.
So, −|x| ≤ 0.
Conclusion:
Range = [ (−∞, 0{]} ]
Question 10. Find the range of [f(x) = 1 / x].
Step-by-Step Solution:
The value [1/x] can never be 0.
It can be positive or negative.
Conclusion:
Range = [ (−∞, 0) ∪ (0, ∞) ]
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