If-fog-x-2x-1-x-and-g-x-5x-2-find-f-x-

Question Number 153583 by pete last updated on 08/Sep/21

If f o g (x) = \frac{2 x - 1}{x} and g (x) = 5 x + 2, find

f (x) .

Commented by otchereabdullai@gmail.com last updated on 08/Sep/21

nice question

Commented by benhamimed last updated on 08/Sep/21

g (x) = y; x = \frac{y - 2}{5}

f (y) = \frac{2 \frac{y - 2}{5} - 1}{\frac{y - 2}{5}} = \frac{2 y - 4 - 5}{y - 2} = \frac{2 y - 9}{y - 2}

f (x) = \frac{2 x - 9}{x - 2}

Commented by pete last updated on 08/Sep/21

Thanks very much sir

Answered by liberty last updated on 08/Sep/21

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

f (5 x + 2) = 2 - \frac{1}{x}

f (\frac{5 x + 2 - 2}{5}) = 2 - \frac{1}{\frac{x - 2}{5}}

f (x) = 2 - \frac{5}{x - 2} = \frac{2 x - 4 - 5}{x - 2} = \frac{2 x - 9}{x - 2}

Commented by pete last updated on 08/Sep/21

Thank you sir

Answered by Rasheed.Sindhi last updated on 08/Sep/21

f o g (x) = \frac{2 x - 1}{x}; g (x) = 5 x + 2; f (x) = ?

f (5 x + 2) = \frac{2 x - 1}{x}

L e t 5 x + 2 = y \Rightarrow x = \frac{y - 2}{5}

f (y) = \frac{2 (\frac{y - 2}{5}) - 1}{\frac{y - 2}{5}} = \frac{2 y - 4 - 5}{y - 2} = \frac{2 y - 9}{y - 2}

y \leftarrow x :

f (x) = \frac{2 x - 9}{x - 2}

Commented by pete last updated on 08/Sep/21

Thank you sir

Answered by amin96 last updated on 08/Sep/21

f o g (x) = 2 - \frac{1}{x} f (x) = \frac{a x + b}{c x + d}

f o g (x) = \frac{5 a x + 2 a + b}{5 c x + 2 c + d} = \frac{2 x - 1}{x} \Rightarrow {\begin{cases} a = \frac{2}{5} \\ 2 a + b = - 1 \\ c = \frac{1}{5} \\ 2 c + d = 0 \end{cases}

b = \frac{9}{5} d = \frac{- 2}{5} a = \frac{2}{5} c = \frac{1}{5}

f (x) = \frac{\frac{2}{5} x - \frac{9}{5}}{\frac{x}{5} - \frac{2}{5}} = \frac{\frac{2 x - 9}{5}}{\frac{x}{5} - \frac{2}{5}} = \frac{2 x - 9}{x - 2}

Answered by amin96 last updated on 08/Sep/21

g (x) = 5 x + 2 \Rightarrow g^{- 1} (x) = \frac{x - 2}{5}

f o f (x) = \frac{2 x - 1}{x} \Rightarrow {f o g o g}^{- 1} (x) = f (x)

f (x) = \frac{2 \frac{(x - 2)}{5} - 1}{\frac{x - 2}{5}} = \frac{2 x - 4 - 5}{x - 2} = \frac{2 x - 9}{x - 2}

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