For x \in R – {0, 1}, let f_{1} \left(\right. x \left.\right) = \frac{1}{x} , f2(x) = 1 – x and f_{3} \left(\right. x \left.\right) = \frac{1}{1 - x} be three given functions. If a function, J(x) satisfies (f2 o J o f1)(x) = f3(x), then J(x) is equal to

Engineering

Mathematics

Introduction to Functions

Even and Odd Functions

functional equation

Question

For x $\in$ R – {0, 1}, let $f_{1} (x) = \frac{1}{x}$ , f₂(x) = 1 – x and $f_{3} (x) = \frac{1}{1 - x}$ be three given functions. If a function, J(x) satisfies (f₂ o J o f₁)(x) = f₃(x), then J(x) is equal to

f₂(x)

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

f₃(x)

f₁(x)

x $\in$ R – {0,1}

$f_{1} (x) = \frac{1}{x}, f_{2} (x) = 1 - x, f_{3} (x) = \frac{1}{1 - x}$

Given $f_{2} (J (f_{1} (x))) = f_{3} (x)$

$1 - J (f_{1} (x)) = f_{3} (x)$

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

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

$J (\frac{1}{x}) = \frac{x}{x - 1} = \frac{1}{1 - \frac{1}{x}}$

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

We are given three functions: $f_{1} (x) = \frac{1}{x}$ , $f_{2} (x) = 1 - x$ , and $f_{3} (x) = \frac{1}{1 - x}$ . We need to find a function J(x) such that (f₂ ∘ J ∘ f₁)(x) = f₃(x). This means we apply f₁ first, then J, then f₂, and the result should equal f₃(x).

Step 1: Understand the composition (f₂ ∘ J ∘ f₁)(x)

This composition means: first apply f₁ to x, then apply J to the result, then apply f₂ to that result. So,

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

Step 2: Substitute the given functions

We know $f_{1} (x) = \frac{1}{x}$ and $f_{2} (x) = 1 - x$ . So,

$f_{2} (J (\frac{1}{x})) = 1 - J (\frac{1}{x}) = f_{3} (x) = \frac{1}{1 - x}$

Step 3: Solve for J(1/x)

From the equation above:

$1 - J (\frac{1}{x}) = \frac{1}{1 - x}$

Now, isolate J(1/x):

$J (\frac{1}{x}) = 1 - \frac{1}{1 - x}$

Simplify the right-hand side:

$J (\frac{1}{x}) = \frac{1 - x}{1 - x} - \frac{1}{1 - x} = \frac{1 - x - 1}{1 - x} = \frac{- x}{1 - x}$

This simplifies to:

$J (\frac{1}{x}) = \frac{x}{x - 1}$

Step 4: Find J(x)

Let $t = \frac{1}{x}$ . Then, $x = \frac{1}{t}$ . Substitute into the equation:

$J (t) = \frac{\frac{1}{t}}{\frac{1}{t} - 1} = \frac{\frac{1}{t}}{\frac{1 - t}{t}} = \frac{1}{t} \times \frac{t}{1 - t} = \frac{1}{1 - t}$

So, $J (t) = \frac{1}{1 - t}$ . Since the variable name doesn't matter, we can write:

$J (x) = \frac{1}{1 - x}$

But note that $f_{3} (x) = \frac{1}{1 - x}$ . Therefore, J(x) = f₃(x).

Final Answer: J(x) = f₃(x)

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