Answer to Question #104052 in Calculus for Akshay

Question

Express the function g: R+-R+ is defined by , g(X) =1/√x+√x, as a composition of three function

Answer 1

$g(x)=\frac{1}{\sqrt{x + \sqrt{x}}}, \quad g: \mathbb{R}^+ \to \mathbb{R}^+$

Let $f(x)=\sqrt{x}, h(x)=\sqrt{x^2+x}, \text{ and } k(x) = \frac{1}{x}.$

Determine $h\circ f.$

$h\circ f = h(f(x)) = \sqrt{(f(x))^2 + f(x)} = \sqrt{(\sqrt{x})^2 + \sqrt{x}} \\ \qquad = \sqrt{x + \sqrt{x}}$

Now, determine $k\circ(h\circ f).$

$k\circ(h\circ f) = k(h(f(x))) = \frac{1}{h(f(x))} = \frac{1}{\sqrt{x + \sqrt{x}} }$

Therefore, $g(x) = k\circ(h\circ f), \text{ where } f(x)=\sqrt{x}, h(x)=\sqrt{x^2+x}, \text{ and } k(x) = \frac{1}{x}.$