Question

Find the sum and product of identity function and the modulus function.

Answer 1

Identity function refers to function which returns the same value when that number was used as its argument, for $f$ being the identity, the equality $f\left( y \right) = y$ for all $y$ whereas argument of a function refers to the value that is provided to obtain result. Here, $y$ is the argument. Modulus function refers to the function which gives absolute value of variables and numbers.

Modulus of a number gives the magnitude of that number represented as $f\left( y \right) = \left| y \right|$ , here if the number is non-negative the magnitude will also be non-negative and if the number is negative then it returns the number $f\left( { - y} \right) = - y$ where $\left| { - y} \right| = y$ is considered as a positive number. A modulus function always returns a positive number whether the input is positive or negative.

Complete step by step answer:
Write an identity function which returns the same value used in the argument as:
$f\left( x \right) = x$
and a modulus function which gives the magnitude of that input
$f\left( x \right) = \left| x \right|$
Hence the sum of the identity function and the modulus function will be:
$f\left( x \right) = x + \left| x \right|$
Which will be equal to $2x$ if input is positive at $x > 0$, and 0 if the input is negative $x \leqslant 0$

{x + \left| x \right| = 2x; x > 0} \\\ {x + \left| x \right| = 0;x \leqslant 0} \end{array}} \right\\}$$ now the product of the identity function and the modulus function will be $$f\left( x \right) = x \times \left| x \right|$$ which will be equal to $${x^2}$$ if input is positive i.e. $$x > 0$$ and $$ - {x^2}$$ if the input is negative $$x \leqslant 0$$ $$f\left( x \right) = \left\\{ {\begin{array}{*{20}{c}} {x \times \left| x \right| = {x^2}; x > 0} \\\ {x \times \left| x \right| = - {x^2}; x \leqslant 0} \end{array}} \right\\}$$ **Note:** Modulus operation can be applied on any real number whose range is $$\left[ {0,\infty } \right]$$ i.e. for all the non-negative numbers. Identity function is also known as identity relation, identity map or identity transformation.