Set, Relations and FunctionsPYQ Dec. 21Question 1968 of 217
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If f(x)=x2−1\displaystyle f(x) = x^2-1 and g(x)=2x+3\displaystyle g(x) = 2x+3, then [fog](3)−gof(−3)]\displaystyle [fog](3)-gof(-3)] is?

Options

A71
B61
C41
D51
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Correct Answer

✅ Option b — 61

All Options:

  • A71
  • B61
  • C41
  • D51

Detailed Solution & Explanation

Given the functions:
f(x)=x2−1f(x) = x^2 - 1
g(x)=2x+3g(x) = 2x + 3
We need to find the value of [f∘g](3)−[g∘f](−3)\displaystyle [f \circ g](3) - [g \circ f](-3).

**Step 1: Calculate [f∘g](3)\displaystyle [f \circ g](3)**
First, find g(3)\displaystyle g(3):
g(3)=2(3)+3=6+3=9g(3) = 2(3) + 3 = 6 + 3 = 9
Now, find f(g(3))=f(9)\displaystyle f(g(3)) = f(9):
f(9)=92−1=81−1=80f(9) = 9^2 - 1 = 81 - 1 = 80
So, [f∘g](3)=80\displaystyle [f \circ g](3) = 80.

**Step 2: Calculate [g∘f](−3)\displaystyle [g \circ f](-3)**
First, find f(−3)\displaystyle f(-3):
f(−3)=(−3)2−1=9−1=8f(-3) = (-3)^2 - 1 = 9 - 1 = 8
Now, find g(f(−3))=g(8)\displaystyle g(f(-3)) = g(8):
g(8)=2(8)+3=16+3=19g(8) = 2(8) + 3 = 16 + 3 = 19
So, [g∘f](−3)=19\displaystyle [g \circ f](-3) = 19.

**Step 3: Calculate the final difference**
[f∘g](3)−[g∘f](−3)=80−19=61[f \circ g](3) - [g \circ f](-3) = 80 - 19 = 61

Hence, **Option B** is the correct answer.

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