A curve C has an equation of $y = f(x)$, where $ f(x) = \frac{4x}{\sqrt{4+x}+\sqrt{4-x}} $, and $ -4\le x\le4. $

Question

This question is Part $A$ of a binomial theorem question. I am unable to continue as I am stuck here. I really appreciate any help.

A curve $C$ has an equation of $y = f(x)$, where

$$ f(x) = \frac{4x}{\sqrt{4+x}+\sqrt{4-x}} $$

and $$ -4\le x\le4. $$

The question is:

• Show that $$ f(x) = k(\sqrt{4+x}-\sqrt{4-x}) $$ for some constant $k$ to be determined.

Answer 1

Multiply by the conjugate. $$f(x) = \dfrac{4x}{\sqrt{4+x}+\sqrt{4-x}}\cdot \dfrac{\sqrt{4+x}-\sqrt{4-x}}{\sqrt{4+x}-\sqrt{4-x}} = \text{?}$$