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Functions: Let Defined Graph Reflection Graph Through Line Domain

JEE Maths question with a full step-by-step solution.

Question
Let f:R→Rf: \mathbb{R} \to \mathbb{R} be defined as f(x)=x3+x+1f(x) = x^3 + x + 1, 1≤x≤21 \leq x \leq 2. The graph of y=g(x)y = g(x) is the reflection of the graph of y=f(x)y = f(x) through the line y=xy = x. If the domain of g(x)g(x) is [a,b][a, b], then ∣a−b∣|a-b| is.
Solution
Answer: 8
Step 1: Find the range of ff f′(x)=3x2+1>0f'(x) = 3x^2+1 > 0, so ff is strictly increasing on [1,2][1,2].
f(1)=1+1+1=3,f(2)=8+2+1=11f(1) = 1+1+1 = 3, \quad f(2) = 8+2+1 = 11
 Step 2: Relate domain of gg to range of ff Since gg is the reflection of ff through y=xy = x, g=f−1g = f^{-1}. The domain of gg equals the range of ff, which is [3,11][3, 11].
a=3,b=11  ⟹  ∣a−b∣=8a = 3, \quad b = 11 \implies |a-b| = 8
 Answer: 8
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