Let the function $f,\ g,\ h$ are defined from the set of real numbers $R$ to $R$ such that $f(x) = x^{2} - 1,\ g(x) = \sqrt{(x^{2} + 1)}$ and $h(x) = \left\{ \begin{array}{r} 0,\ if\ x

Question

Let the function $f,\ g,\ h$ are defined from the set of real numbers $R$ to $R$ such that $f(x) = x^{2} - 1,\ g(x) = \sqrt{(x^{2} + 1)}$ and $h(x) = \left\{ \begin{array}{r} 0,\ if\ x < 0 \ x,\ if\ x \geq 0 \end{array} ight.\ ,$ then $ho(fog)(x)$ is defined by

Dr. Abubacker · Accepted Answer

(b)
$ho(fog)(x) = hof\left\{ g(x) \right\}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  = hof\{\sqrt{{(x}^{2} + 1})\}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  = h\{\left( \sqrt{x^{2} + 1} \right)^{2} - 1\}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  = h\{ x^{2} + 1 - 1\}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  = h\{ x^{2}\} = x^{2}$

Q. Let the function $f,\ g,\ h$ are defined from the set of real numbers $R$ to $R$ such that $f(x) = x^{2} - 1,\ g(x) = \sqrt{(x^{2} + 1)}$ and $h(x) = \left\{ \begin{array}{r} 0,\ if\ x < 0 \\ x,\ if\ x \geq 0 \end{array} \right.\ ,$ then $ho(fog)(x)$ is defined by

Explanation

Reviewed by Dr. Abubacker

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