For the following two (02) items: Consider the function f(x) = (x)/(1 - x) (x > 0 , x ≠ 1 ). What is (f(x))/(f(x+1)) equal to?

Correct answer: A. -f(x^2) Given f(x) = (x)/(1 - x) Substitute x+1 for x to find f(x+1) : f(x+1) = (x+1)/(1 - (x+1)) = (x+1)/(-x) Now, calculate the ratio (f(x))/(f(x+1)) : (f(x))/(f(x+1)) = ((x)/(1 - x))/((x+1)/(-x)) (f(x))/(f(x+1)) = (x)/(1 - x) × (-x)/(x+1) = (-x^2)/(1 - x^2) Since f(x^2) = (x^2)/(1 - x^2) , we get: (f(x))/(f(x+1)) = -f(x^2) Answer: -f(x^2)

NDA 2025

Exam: NDA
Year: 2025
Subject: Mathematics
Chapter: Functions

NDA 2025 · Mathematics · Functions

For the following two (02) items: Consider the function f(x) =...

For the following two (02) items:
Consider the function

f(x) = \dfrac{x}{1 - x}

(x > 0

,

x \neq 1

).<br><br>What is \dfrac{f(x)}{f(x+1)}

equal to?

Options

A $-f(x^2)$
B $-f(\sqrt{x})$
C $f(x^2)$
D $f(x - 1)$

Correct answer

A. $-f(x^2)$

Step-by-step solution

Given

f(x) = \dfrac{x}{1 - x}

Substitute

x+1

for

x

to find

f(x+1)

f(x+1) = \dfrac{x+1}{1 - (x+1)} = \dfrac{x+1}{-x}

Now, calculate the ratio

\dfrac{f(x)}{f(x+1)}

\dfrac{f(x)}{f(x+1)} = \dfrac{\dfrac{x}{1 - x}}{\dfrac{x+1}{-x}}

\dfrac{f(x)}{f(x+1)} = \dfrac{x}{1 - x} \times \dfrac{-x}{x+1} = \dfrac{-x^2}{1 - x^2}

Since

f(x^2) = \dfrac{x^2}{1 - x^2}

, we get:

\dfrac{f(x)}{f(x+1)} = -f(x^2)

Answer:

-f(x^2)

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