Q.19 If φ(x) = x² and ψ(x) = 2x, then ψ(φ(x)) is
The correct answer is (A) 2x². ψ(φ(x)) represents the composition of functions where φ(x) is evaluated first, then ψ applied to that result. This simplifies directly to 2(x²), matching option (A).
Solution Steps
Start with the given functions: φ(x) = x² and ψ(x) = 2x.
Composition ψ(φ(x)) means substitute φ(x) into ψ: ψ(φ(x)) = ψ(x²) = 2(x²) = 2x².
Verify with x=1: φ(1)=1, ψ(1)=2(1)=2; direct 2(1)²=2. Matches perfectly.
Option Analysis
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(A) 2x²: Correct. Direct substitution ψ(φ(x)) = 2(x²).
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(B) x²: Incorrect. Equals φ(x) alone, ignores outer ψ multiplication by 2.
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(C) 2²x: Incorrect. Equals 4x, misinterprets as ψ(x²)= (2²)x instead of 2(x²).
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(D) x²x: Incorrect. Equals x³ (x·x²), confuses multiplication with composition.
In ψ(φ(x)) composition of functions problems like “If φ(x) = x² and ψ(x) = 2x, then ψ(φ(x)) is,” students often confuse order or notation. This MCQ tests core algebra: apply inner function φ first, then outer ψ.
Composition means ψ(φ(x)) = ψ(of x²) = 2x², eliminating distractors like x³ or 4x.
| Option | Expression | Why Wrong/Correct |
|---|---|---|
| (A) | 2x² | ψ(x²)=2(x²). Right! |
| (B) | x² | Just φ(x). Misses ψ. |
| (C) | 2²x=4x | Wrong exponent order. |
| (D) | x²x=x³ | Multiplication, not composition. |
ψ(φ(x)) composition of functions appears in CSIR NET under mathematical methods. Practice: fog(x)=f(g(x)), never reverse.
