Question:
A function f : R+ → R (where R+ is the set of all non-negative real numbers) is defined by f(x) = 4x + 3. Then f is:
(A) one–one but not onto
(B) onto but not one–one
(C) both one–one and onto
(D) neither one–one nor onto
Solution:
One–one:
Let f(x₁) = f(x₂).
Then 4x₁ + 3 = 4x₂ + 3 ⇒ x₁ = x₂.
Hence, f is one–one.
Onto:
For f to be onto, for every y ∈ R there must exist x ∈ R+ such that
y = 4x + 3 ⇒ x = (y − 3)/4.
Since x ≥ 0, we must have y ≥ 3.
Thus, values y < 3 are not obtained.
Hence, f is not onto R.
Conclusion:
The function is one–one but not onto.
Correct option: (A)
AI Tutor Explanation
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Step-by-Step Solution
Check for one-one:
Assume \(f(x_1) = f(x_2)\) for some \(x_1, x_2 \in R_{+}\).
Then, \(4x_1 + 3 = 4x_2 + 3\).
Subtracting 3 from both sides, we get \(4x_1 = 4x_2\).
Dividing by 4, we get \(x_1 = x_2\).
Since \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\), the function is one-one.
Check for onto:
For a function to be onto, its range must be equal to its codomain. The codomain is given as \(R\), the set of all real numbers.
Let \(y = f(x) = 4x + 3\). We need to check if for every \(y \in R\), there exists an \(x \in R_{+}\) such that \(y = 4x + 3\).
Solving for \(x\), we get \(x = \frac{y - 3}{4}\).
Since \(x \in R_{+}\), we must have \(x \geq 0\).
Therefore, \(\frac{y - 3}{4} \geq 0\), which implies \(y - 3 \geq 0\), so \(y \geq 3\).
This means that the range of \(f\) is \([3, \infty)\), which is not equal to the codomain \(R\).
Therefore, the function is not onto.
Conclusion:
The function is one-one but not onto.
Correct Answer: one-one but not onto
AI Suggestion: Option A
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Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because the student needs to apply their understanding of one-one and onto functions to a specific function definition to determine its properties.
Knowledge Dimension:CONCEPTUAL
Justification:The question requires understanding the concepts of one-one and onto functions, rather than just recalling facts or following a specific procedure.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of the definitions and properties of one-one and onto functions, which is a core concept covered in the textbook.