A single formula that gives f(x) for all x>0, where f(x)={2+x,0≤x<23x−2,x≥2 is

f(x)={2+x,0≤x<23x−2,x≥2f(x)=|x−2|+2xif 0≤x<2 then |x−2|=−x+2f(x)=−x+2+2x=x+2if x≥2 then |x−2|=x−2f(x)=x−2+2x=3x−2∴ f(x)={2+x,0≤x<23x−2,x≥2

A single formula that gives f(x) for all x>0, where f(x)={2+x,0≤x<23x−2,x≥2 is

f(x)={2+x,0≤x<23x−2,x≥2f(x)=|x−2|+2xif 0≤x<2 then |x−2|=−x+2f(x)=−x+2+2x=x+2if x≥2 then |x−2|=x−2f(x)=x−2+2x=3x−2∴ f(x)={2+x,0≤x<23x−2,x≥2

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$A \sin g l e f o r m u l a t h a t g i v e s f (x) f o r a l l x > 0, w h e r e f (x) = {\begin{cases} 2 + x, 0 \leq x < 2 \\ 3 x - 2, x \geq 2 \end{cases} i s$

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$f (x) = | x - 2 | + 2 x$

$f (x) = | 2 x - 1 | + x$

$f (x) = | 3 x - 1 | - 3$

$f (x) = | 3 x - 2 | + 1$

answer is A.

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Detailed Solution

$\begin{array}{l} f (x) = {\begin{cases} 2 + x, 0 \leq x < 2 \\ 3 x - 2, x \geq 2 \end{cases} \\ f (x) = | x - 2 | + 2 x \\ i f 0 \leq x < 2 t h e n | x - 2 | = - x + 2 \\ f (x) = - x + 2 + 2 x = x + 2 \\ i f x \geq 2 t h e n | x - 2 | = x - 2 \\ f (x) = x - 2 + 2 x = 3 x - 2 \\ ∴ f (x) = {\begin{cases} 2 + x, 0 \leq x < 2 \\ 3 x - 2, x \geq 2 \end{cases} \end{array}$