Set, Relations and FunctionsMTP May 18Question 1982 of 217
All Questions

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.

Options

Ax2(2x2+1)\displaystyle x^2(2x^2+1)
Bx2\displaystyle x^2
C2x4+1\displaystyle 2x^4+1
D(2x2+1)2\displaystyle (2x^2+1)^2
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Correct Answer

✅ Option c — 2x4+1\displaystyle 2x^4+1

All Options:

  • Ax2(2x2+1)\displaystyle x^2(2x^2+1)
  • Bx2\displaystyle x^2
  • C2x4+1\displaystyle 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** (2x4+1\displaystyle 2x^4+1) as the correct answer. This indicates the authors made a composition order error, accidentally computing g∘f\displaystyle g \circ f instead of f∘g\displaystyle f \circ g.

Hence, **Option C** is the correct answer (according to the textbook key), but the mathematically proven correct value for f∘g\displaystyle f \circ g is (2x2+1)2\displaystyle (2x^2+1)^2 (**Option D**).

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