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Standard Formulae - 1

given f(x)={...

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given f(x)=\{ 4x-2; x《=1 and g(x)=\{ x+1; -1《=x《2 x^2; 1《x《=2 2x+3; 2《= x《=3 The range of g(f(x)

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f (x) = {\begin{matrix} x^{2} - 4 x + 3, & x < 2 x - 4, & x \geq 2 \end{matrix}

g (x) = {\begin{matrix} x - 3, & x < 3 x^{2} + 2 x + 2, & x \geq 3 \end{matrix}

\frac{f}{g} \cdot (1)

f (x) = {\begin{matrix} x + 1, & x > 0 2 - x, & x \leq 0 \end{matrix}

g (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} x + 3, & x < 1 x^{2} - 2 x - 2, & 1 \leq x < 2 x - 5, & x \geq 2 \end{matrix}

, then the value of

lim x \to 0 g (f (x))

f (x) = {\begin{matrix} 2 - x, & x \leq 0 x + 1, & x > 0 \end{matrix}

g (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} x + 3, & x < 1 x^{2} - 2 x - 2, & 1 \leq x < 2 x - 5, & x \geq 2 \end{matrix}

g (f (x))

is continuous at

x = 0

, then the value of

g (f (0))

f (x) = \frac{3}{x - 2} - \frac{1}{x + 3}

g (x) = \frac{x^{2} - 4 x + 19}{x^{2} + x - 6}

f (x) = g (x)

x =

f (x) = {\begin{matrix} sin x; x \neq n π, n = 0, \pm 1, \pm 2, \pm 3..... 2; o t h e r w i s e \end{matrix}

g (x) = {\begin{matrix} x^{2} + 1; x \neq 0 4; x = 0 \end{matrix} .

{lim}_{x \to 0} g (f (x))

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