Sets, Relations and FunctionsMCQMTP May 18Question 1982 of 222
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Find f∘g\displaystyle f \circ g for the functions f(x)=x2\displaystyle f(x) = x^2, g(x)=2x2+1\displaystyle g(x) = 2x^2+1.

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option c — $2x^4+1$

All Options:

  • Ax2(2x2+1)\displaystyle x^2(2x^2+1)
  • Bx2\displaystyle x^2
  • C$2x^4+1$
  • D(2x2+1)2\displaystyle (2x^2+1)^2

Detailed Solution & Explanation

Let us analyze both orders of composition for the given functions f(x)=x2\displaystyle f(x) = x^2 and g(x)=2x2+1\displaystyle g(x) = 2x^2+1:

**1. Mathematical Derivation of f∘g\displaystyle f \circ g (Requested):**
(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))
Substitute g(x)\displaystyle g(x) into f(x)\displaystyle f(x):
f(g(x))=(2x2+1)2f(g(x)) = (2x^2 + 1)^2
This corresponds to **Option D**.

**2. Mathematical Derivation of g∘f\displaystyle g \circ f (Alternative):**
(g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x))
Substitute f(x)\displaystyle f(x) into g(x)\displaystyle g(x):
g(f(x))=2(f(x))2+1=2(x2)2+1=2x4+1g(f(x)) = 2(f(x))^2 + 1 = 2(x^2)^2 + 1 = 2x^4 + 1
This corresponds to **Option C**.

**Discrepancy & Design Error:**
The question asks for f∘g\displaystyle f \circ g (which mathematically yields (2x2+1)2\displaystyle (2x^2+1)^2, Option D). However, the textbook answer key lists **Option C** ($2x^4+1$) as the correct answer. This indicates the authors made a composition order error, accidentally computing $g \circ finsteadof\displaystyle instead off \circ g.<br><br>Hence,∗∗OptionC∗∗isthecorrectanswer(accordingtothetextbookkey),butthemathematicallyprovencorrectvaluefor\displaystyle .<br><br>Hence, **Option C** is the correct answer (according to the textbook key), but the mathematically proven correct value forf \circ gis\displaystyle is(2x^2+1)^2$ (**Option D**).

About This Chapter: Sets, Relations and Functions

Paper

Paper 3: Quantitative Aptitude

Weightage

3-5 Marks

Key Topics

Sets, Relations, Functions

This chapter covers Sets, Relations, Functions and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 3-5 Marks weightage. Focus on understanding core concepts rather than memorizing.

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