Question

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

Answer 1

Hint: 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 \[x \leqslant 0\]
 \[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  {x + \left| x \right| = 2x; x > 0} \\
  {x + \left| x \right| = 0;x \leqslant 0}
\end{array}} \right\}\]
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.