Set, Relations and FunctionsMTP Oct 21Question 1981 of 217
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Find g∘f\displaystyle g \circ f for the functions f(x)=x\displaystyle f(x) = \sqrt{x}, g(x)=2x2+1\displaystyle g(x) = 2x^2+1.

Options

A2x2+1\displaystyle 2x^2+1
B2x+1\displaystyle 2x+1
C(2x2+1)(x)\displaystyle (2x^2+1)(\sqrt{x})
D10\displaystyle \sqrt{10}
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Correct Answer

✅ Option a — 2x2+1\displaystyle 2x^2+1

All Options:

  • A2x2+1\displaystyle 2x^2+1
  • B2x+1\displaystyle 2x+1
  • C(2x2+1)(x)\displaystyle (2x^2+1)(\sqrt{x})
  • D10\displaystyle \sqrt{10}

Detailed Solution & Explanation

To find the composite function (g∘f)(x)=g(f(x))\displaystyle (g \circ f)(x) = g(f(x)), we substitute the expression for f(x)\displaystyle f(x) into g(x)\displaystyle g(x):
Given:
f(x)=xf(x) = \sqrt{x}
g(x)=2x2+1g(x) = 2x^2 + 1

Substitute f(x)\displaystyle f(x) in place of x\displaystyle x in g(x)\displaystyle g(x):
g(f(x))=2(x)2+1g(f(x)) = 2(\sqrt{x})^2 + 1
Since (x)2=x\displaystyle (\sqrt{x})^2 = x (for x≥0\displaystyle x \ge 0), this simplifies to:
g(f(x))=2x+1g(f(x)) = 2x + 1
This corresponds to **Option B**.

**Discrepancy & Typographical Error:**
Mathematically, the composite function simplifies to 2x+1\displaystyle 2x+1, which corresponds to **Option B**. However, the mock exam answer key (MTP Oct 21) contains a typographical error, designating **Option A** (2x2+1\displaystyle 2x^2+1) as correct. The mathematically correct expression is 2x+1\displaystyle 2x+1 (Option B).

Hence, **Option A** is the correct answer (according to the textbook key), but the mathematically proven correct value is 2x+1\displaystyle 2x+1 (**Option B**).

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