Set, Relations and FunctionsMTP June 24 Series IIQuestion 2004 of 217
All Questions

Given the function f(x)=(2x+3)\displaystyle f(x)=(2x+3), then the value of f(2x)−2f(x)+3\displaystyle f(2x)-2f(x)+3 will be:

Options

A3\displaystyle 3
B2\displaystyle 2
C1\displaystyle 1
D0\displaystyle 0
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Correct Answer

✅ Option d — 0\displaystyle 0

All Options:

  • A3\displaystyle 3
  • B2\displaystyle 2
  • C1\displaystyle 1
  • D0\displaystyle 0

Detailed Solution & Explanation

Given the function:
f(x)=2x+3f(x) = 2x + 3
We need to evaluate the expression f(2x)−2f(x)+3\displaystyle f(2x) - 2f(x) + 3.

**Step 1: Compute f(2x)\displaystyle f(2x)** by substituting 2x\displaystyle 2x in place of x\displaystyle x:
f(2x)=2(2x)+3=4x+3f(2x) = 2(2x) + 3 = 4x + 3

**Step 2: Compute 2f(x)\displaystyle 2f(x)** by multiplying the entire function f(x)\displaystyle f(x) by 2\displaystyle 2:
2f(x)=2(2x+3)=4x+62f(x) = 2(2x + 3) = 4x + 6

**Step 3: Substitute these into the given expression and simplify**:
f(2x)−2f(x)+3=(4x+3)−(4x+6)+3f(2x) - 2f(x) + 3 = (4x + 3) - (4x + 6) + 3
Remove the parentheses:
=4x+3−4x−6+3= 4x + 3 - 4x - 6 + 3
Group the variable terms and constant terms:
=(4x−4x)+(3−6+3)= (4x - 4x) + (3 - 6 + 3)$
=0+0=0= 0 + 0 = 0

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

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