Functions
The function f: R → R defined by f(x) = x³ + 5 is
Select the correct option:
Solution
Both one-one and onto
- Test for One-one (Injection): f(x1)=f(x2)⟹x13+5=x23+5⟹x13=x23. Since cube roots of real numbers are unique, x1=x2. Thus, f is one-one.
- Test for Onto (Surjection): We need to see if every y∈R has a pre-image.
- y=x3+5⟹x=3y−5.
- Since the cube root of any real number is always real, x∈R for all y. Thus, f is onto.
- Conclusion: f is a bijection.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- functions
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
Both one-one and onto
- Test for One-one (Injection): f(x1)=f(x2)⟹x13+5=x23+5⟹x13=x23. Since cube roots of real numbers are unique, x1=x2. Thus, f is one-one.
- Test for Onto (Surjection): We need to see if every y∈R has a pre-image.
- y=x3+5⟹x=3y−5.
- Since the cube root of any real number is always real, x∈R for all y. Thus, f is onto.
- Conclusion: f is a bijection.
This medium difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of functions. It appeared in the 2025 exam.
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