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How do you express the following function, \[f\left( x \right)\] as a composition of two functions f and g given \[f\left( x \right)=\dfrac{{{x}^{2}}}{{{x}^{2}}+4}\]?

Last updated date: 13th Jun 2024
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Hint: For this question we are asked to find the two functions that could form that composite function of each other. Multiple answers are possible to these kinds of questions so for the questions of those kinds the explanation or the solution will be as follows.

Complete step-by-step solution:
\[\Rightarrow f\left( x \right)=\dfrac{{{x}^{{2}}}}{x+4}\]
In mathematics function composition is an operation that takes two functions f and g and produces a function h such that \[h\left( x \right)\] which will be equal to g. In this operation the function g is applied to the result of applying the function f to x.
\[\Rightarrow g\left( x \right)={{x}^{2}}\]
\[\Rightarrow f\left( x \right)=h\left( g\left( x \right) \right)\]
Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function of the two functions will be denoted by
\[\Rightarrow g\left( f\left( x \right) \right)=z\]
Defined by function, that is the function will be valid only in the domain of x which belongs to the real numbers.

Note: we the people who are doing the problems of these kinds must have by maximum knowledge of the concept of functions topic mainly composite of a function. These questions of kind will be generally testing the person's knowledge and depth of the person in the function concept because we use only a simple assuming because these questions can have many solutions example this question \[f\left( x \right)=\dfrac{{{x}^{2}}}{{{x}^{2}}+4}\]
Can have many solutions like \[g\left( x \right)={{x}^{2}}\],\[f\left( x \right)=h\left( g\left( x \right) \right)\] and many of these kind. People who are doing questions of this kind must have deep knowledge in the concept of functions.