Functions Question - None

Question

If f\left( x \right) = \frac{4x}{4x + 2}, then f\left( x \right) + f\left( 1 - x \right) is equal to-

Accepted Answer

Key Idea: Break domain at x=2 where [x] changes.

Step 1: $x\in(1,2)$ : $$f(x)=x-1$$ , range (0,1)

Step 2: $x\in[2,3)$ : $$f(x)=1+(x-2)^2$$ , range [1,2)

Step 3: Inverting: $f^{-1}(x) = \begin{cases}x+1, & x\in(0,1)\ 2+\sqrt{x-1}, & x\in[1,2)\end{cases}$

Trap: Do not use

2-\sqrt{x-1}

in second branch — that violates

x\ge 2

.

Key Concept: Piecewise GIF/fractional-part inverse via interval splitting

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