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Composite Function

For x∈ℝ - 0, ...

Question

For $x \in R - {0, 1},$ let $f_{1} (x) = \frac{1}{x}$ , $f_{2} (x) = 1 - x$ and $f_{3} (x) = \frac{1}{1 - x}$ be three given functions. If a function, $J (x)$ satisfies $(f_{2} o J o f_{1}) (x) = f_{3} (x)$ , then $J (x)$ is equal to :

f_{1} (x)

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f_{2} (x)

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f_{3} (x)

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\frac{1}{x} \cdot f_{3} (x)

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Solution

The correct option is C $f_{3} (x)$
Given :
$f_{1} (x) = \frac{1}{x},$
$f_{2} (x) = 1 - x,$
$f_{3} (x) = \frac{1}{1 - x}$
$(f_{2} o J o f_{1}) (x) = f_{3} (x) . . . (1)$

Substituting the values of $f_{1} (x)$ and $f_{3} (x)$ in equation $(1),$ we will get

$(f_{2} o J) (\frac{1}{x}) = \frac{1}{1 - x}$
$\Rightarrow f_{2} (J (\frac{1}{x})) = \frac{1}{1 - x}$

Substituting $J (\frac{1}{x})$ in the function $f_{2} (x)$ we will get,
$1 - J (\frac{1}{x}) = \frac{1}{1 - x}$
$\Rightarrow 1 - \frac{1}{1 - x} = J (\frac{1}{x})$
$\Rightarrow 1 - \frac{\frac{1}{x}}{\frac{1}{x} - 1} = J (\frac{1}{x})$

Putting $\frac{1}{x} = t$ ,
$\Rightarrow 1 - \frac{t}{t - 1} = J (t)$
$\Rightarrow \frac{t - 1 - t}{t - 1} = J (t)$
$\Rightarrow \frac{- 1}{t - 1} = J (t)$
$∴ J (t) = \frac{1}{1 - t}$
Replacing $t \to x$
So , $J (x) = \frac{1}{1 - x} = f_{3} (x)$

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Similar questions

Q. For

x ϵ R - {0, 1}

, let

f_{1} (x) = \frac{1}{x}, f_{2} (x) = 1 - x

and

f_{3} (x) = \frac{1}{1 - x}

be three given functions. If a function,

J (x)

satisfies

(f_{2}^{\circ} J^{\circ} f_{1}) (x) = f_{3} (x)

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Q. For

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f_{1} (x) = \frac{1}{x}

f_{2} (x) = 1 - x

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f_{3} (x) = \frac{1}{1 - x}

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Q. If

f_{1} (x) = 2^{f_{2} (x)} .

Where

f_{2} (x) = 2012^{f_{3} (x)} .

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f_{3} (x) = {(\frac{1}{2013})}^{f_{4} (x)} .

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f_{4} (x) = {log}_{2013} {log}_{x} 2012,

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Which of the following is necessarily true?

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Composite Function

Standard XII Mathematics

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For x∈R−{0,1}, let f1(x)=1x, f2(x)=1−x and f3(x)=11−x be three given functions. If a function, J(x) satisfies (f2oJof1)(x)=f3(x), then J(x) is equal to :

For $x \in R - {0, 1},$ let $f_{1} (x) = \frac{1}{x}$ , $f_{2} (x) = 1 - x$ and $f_{3} (x) = \frac{1}{1 - x}$ be three given functions. If a function, $J (x)$ satisfies $(f_{2} o J o f_{1}) (x) = f_{3} (x)$ , then $J (x)$ is equal to :