A functionis defined by f(x) = [x+1], where [x] the greatest integer function, is:a)onto but not one-oneb)Neither one-one nor ontoc)One-one but not ontod)One-one and ontoCorrect answer is option 'B'. Can you explain this answer?

Question

A functionis defined by f(x) = [x+1], where [x] the greatest integer function, is:&#8203;a)onto but not one-oneb)Neither one-one nor ontoc)One-one but not ontod)One-one and ontoCorrect answer is option 'B'. Can you explain this answer?

Answer

Answer

Correct Answer :- b

Explanation : f: R → R is given by,

f(x) = [x+1]

It is seen that f(1.2) = [1.2] = 1, f(1.9) = [1.9] = 1.

∴ f(1.2) = f(1.9), but 1.2 ≠ 1.9.

∴ f is not one-one.

Now, consider 0.7 ∈ R.

It is known that f(x) = [x] is always an integer. Thus, there does not exist any element x ∈ R such that f(x) = 0.7.

∴ f is not onto.

Hence, the greatest integer function is neither one-one nor onto.

Answer

Correct Answer :- b

Explanation : f: R → R is given by,

f(x) = [x+1]

It is seen that f(1.2) = [1.2] = 1, f(1.9) = [1.9] = 1.

∴ f(1.2) = f(1.9), but 1.2 ≠ 1.9.

∴ f is not one-one.

Now, consider 0.7 ∈ R.

It is known that f(x) = [x] is always an integer. Thus, there does not exist any element x ∈ R such that f(x) = 0.7.

∴ f is not onto.

Hence, the greatest integer function is neither one-one nor onto.

Answer

Understanding the Function f(x) = [x + 1]
The function $ f: \mathbb{R} \rightarrow \mathbb{R} $ is defined as $ f(x) = [x + 1] $, where $[x]$ denotes the greatest integer less than or equal to $ x $. To analyze this function, we must determine whether it is one-one (injective) or onto (surjective).

One-One (Injective) Analysis
- A function is one-one if different inputs yield different outputs.
- For $ f(x) $:
- If $ x_1 $ and $ x_2 $ are two distinct real numbers such that $ x_1 < x_2='' $,='' it='' is='' possible='' that='' $='' f(x_1)='f(x_2)' />
- For example, \( f(0.5) = [1.5] = 1 $ and $ f(0.9) = [1.9] = 1 $.
- This demonstrates that $ f(x) $ is **not one-one** since multiple inputs can produce the same output.

Onto (Surjective) Analysis
- A function is onto if every possible output in the codomain is achieved by some input.
- The output of $ f(x) $ is restricted to integer values, specifically all integers $ n $ such that $ n \in \mathbb{Z} $.
- However, the function cannot produce non-integer values (e.g., $ 0.5 $ or $ 2.3 $).
- Therefore, there are real numbers in the codomain $ \mathbb{R} $ that are not images of any $ x \in \mathbb{R} $.

Conclusion
- Since $ f(x) $ is neither one-one nor onto, the correct answer is option **b) Neither one-one nor onto**.
This analysis provides clarity on the characteristics of the function $ f(x) = [x + 1] $.

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