Physics and Mathematics
Find the Domain and range of the function
Key Idea
- Domain: All possible values of [x] for which the function is defined.
- Range: All possible values of [y = f(x)] that the function can produce.
In most problems:
- First, find the domain
- Then, using the domain, find the range
Basic Rules to Remember
While finding Domain:
- Denominator ≠0
- Quantity inside [√ ≥ 0]
- Quantity inside [log > 0]
While finding Range:
- Use known results (square ≥ 0, modulus ≥ 0, etc.)
- Convert [y = f(x)] and analyze possible values of y
- Use inequalities wherever needed
Examples with Solutions
Example 1. Find the domain and range of [f(x) = x²].
Solution:
- x² is defined for all real x.
- x² ≥ 0 for all x.
Domain: [ (−∞, ∞) ]
Range: [ {[}0, ∞) ]
Example 2. Find the domain and range of [f(x) = √(x − 1)].
Solution:
- For square root, [x − 1 ≥ 0] ⇒ [x ≥ 1]
- √(x − 1) ≥ 0
Domain: [ {[}1, ∞) ]
Range: [ {[}0, ∞) ]
Example 3. Find the domain and range of [f(x) = 1 / (x − 2)].
Solution:
- Denominator cannot be zero ⇒ [x ≠2]
- Value of function can never be zero
Domain: [ (−∞, 2) ∪ (2, ∞) ]
Range: [ (−∞, 0) ∪ (0, ∞) ]
Example 4. Find the domain and range of [f(x) = |x|].
Solution:
- Modulus is defined for all real x
- |x| ≥ 0
Domain: [ (−∞, ∞) ]
Range: [ {[}0, ∞) ]
Example 5. Find the domain and range of [f(x) = 3].
Solution:
- Constant function is defined for all real x
- Output is always 3
Domain: [ (−∞, ∞) ]
Range: [{3}]
Practice Questions with Step-by-Step Solutions
Question 1. Find the domain and range of [f(x) = x + 5].
Step-by-Step Solution:
Linear function ⇒ defined for all real x
Linear functions take all real values
Conclusion:
Domain = [ (−∞, ∞) ]
Range = [ (−∞, ∞) ]
Question 2. Find the domain and range of [f(x) = x² + 4].
Step-by-Step Solution:
[x²] is defined for all real x
Since [x² ≥ 0], [x² + 4 ≥ 4]
Conclusion:
Domain = [ (−∞, ∞) ]
Range = [ {[}4, ∞) ]
Question 3. Find the domain and range of [f(x) = √(5 − x)].
Step-by-Step Solution:
For square root, [5 − x ≥ 0] ⇒ [x ≤ 5]
Square root gives non-negative values
Conclusion:
Domain = [ (−∞, 5] ]
Range = [ {[}0, ∞) ]
Question 4. Find the domain and range of [f(x) = 1 / x].
Step-by-Step Solution:
Denominator cannot be zero ⇒ [x ≠0]
Value of [1/x] can never be zero
Conclusion:
Domain = [ (−∞, 0) ∪ (0, ∞) ]
Range = [ (−∞, 0) ∪ (0, ∞) ]
Question 5. Find the domain and range of [f(x) = |x − 2|].
Step-by-Step Solution:
Modulus is defined for all real x
Modulus gives non-negative values
Conclusion:
Domain = [ (−∞, ∞) ]
Range = [ {[}0, ∞) ]
Question 6. Find the domain and range of [f(x) = √(x²)].
Step-by-Step Solution:
[√(x²) = |x|]
[|x| ≥ 0] for all x
Conclusion:
Domain = [ (−∞, ∞) ]
Range = [ {[}0, ∞) ]
Question 7. Find the domain and range of [f(x) = x / (x − 1)].
Step-by-Step Solution:
Denominator ≠0 ⇒ [x ≠1]
Function can take all real values except 1
Conclusion:
Domain = [ (−∞, 1) ∪ (1, ∞) ]
Range = [ (−∞, 1) ∪ (1, ∞) ]
Question 8. Find the domain and range of [f(x) = √(x² − 4)].
Step-by-Step Solution:
For square root, the expression inside must be ≥ 0.
[x² − 4 ≥ 0]
[x² ≥ 4] ⇒ [x ≤ −2] or [x ≥ 2]
Domain:
[ (−∞, −2{]} ∪ {[}2, ∞) ]
Square root always gives non-negative values.
Range:
[ {[}0, ∞) ]
Conclusion:
Domain = [ (−∞, −2{]} ∪ {[}2, ∞) ]
Range = [ {[}0, ∞) ]
Question 9. Find the domain and range of [f(x) = |x| + 3].
Step-by-Step Solution:
Modulus function is defined for all real x.
[|x| ≥ 0] for all x.
Therefore, [|x| + 3 ≥ 3].
Conclusion:
Domain = [ (−∞, ∞) ]
Range = [ {[}3, ∞) ]
Question 10. Find the domain and range of [f(x) = 1 / (x² + 1)].
Step-by-Step Solution:
Since x² ≥ 0,
[x² + 1 ≥ 1] for all real x.
Denominator is never zero.
Domain:
[ (−∞, ∞) ]
Since [x² + 1 ≥ 1],
[1 / (x² + 1) ≤ 1] and is always positive.
Range:
[ (0, 1{]} ]
Conclusion:
Domain = [ (−∞, ∞) ]
Range = [ (0, 1{]} ]
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