CUET · MATHS · PYQ PAPER 2023

Let \(f: R \rightarrow R\) defined by \(f(x)=2 x^3-7\) for \(x \in R\). Then :
(A) \(f\) is one-one function
(B) \(f\) is many to one function
(C) \(f\) is bijective function
(D) \(f\) is into function
Choose the correct answer from the options given below:

A (A) and (D) only
B (B) and (D) only
C (B) and (C) only
D (A) and (C) only

Verified Solution

Answer & Solution

Correct Answer

(D) (A) and (C) only

Step-by-step Solution

Detailed explanation

\(f(x_1) = f(x_2) \Rightarrow 2x_1^3 - 7 = 2x_2^3 - 7\) \(x_1^3 = x_2^3 \Rightarrow x_1 = x_2\) \(f\) is one-one. For any \(y \in R\), let \(y = 2x^3 - 7 \Rightarrow x = \sqrt[3]{\frac{y+7}{2}}\). Since \(\sqrt[3]{\frac{y+7}{2}} \in R\) for all \(y \in R\), \(f\) is onto. As…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let \(f: R \rightarrow R\) defined by \(f(x)=2 x^3-7\) for \(x \in R\). Then :(A) \(f\) is one-one function(B) \(f\) is many to one function(C) \(f\) is bijective function(D) \(f\) is into functionChoose the correct answer from the options given below:

Step-by-step Solution

Let \(f: R \rightarrow R\) defined by \(f(x)=2 x^3-7\) for \(x \in R\). Then :
(A) \(f\) is one-one function
(B) \(f\) is many to one function
(C) \(f\) is bijective function
(D) \(f\) is into function
Choose the correct answer from the options given below: