Question

How do you write the combined function as a composition of several functions if $f\left( g\left( x \right) \right)=\sqrt{1-{{x}^{2}}}+2$ ?

Answer 1

In this question we have been asked to write the combined function as a composition of several functions if $f\left( g\left( x \right) \right)=\sqrt{1-{{x}^{2}}}+2$ . For doing that let us assume $g\left( x \right)=1-{{x}^{2}}$ . Here in this type of function the range or output of one function $g$ will act as domain or input of the other function $f$ .

Complete step-by-step solution:
Now considering from the question we have been asked to write the combined function as a composition of several functions if $f\left( g\left( x \right) \right)=\sqrt{1-{{x}^{2}}}+2$ .
For doing that let us assume $g\left( x \right)=1-{{x}^{2}}$ .
Here in this type of function the range or output of one function $g$ will act as domain or input of the other function $f$ .
As per our assumption we can simply write the given expression as $f\left( 1-{{x}^{2}} \right)=\sqrt{1-{{x}^{2}}}+2$ .
Let us assume that $g\left( x \right)=y\Rightarrow 1-{{x}^{2}}$ that leads to $f\left( y \right)=\sqrt{y}+2$ .
Hence we can say that when $f\left( g\left( x \right) \right)=\sqrt{1-{{x}^{2}}}+2$ and $g\left( x \right)=1-{{x}^{2}}$ then $f\left( x \right)=\sqrt{x}+2$.

Note: During the process of answering questions of this type we should be careful with our assumptions that we make and simplifications that we perform. Our value of $f\left( x \right)$ depends on the assumed value of $g\left( x \right)$ . If we assume $g\left( x \right)=\sqrt{1-{{x}^{2}}}$ then we will have $f\left( x \right)=x+2$ . Hence we can say that clearly both the functions are interdependent. This is a very simple and easy question. The only thing to be done carefully is the assumption. Very few mistakes are possible in questions of this type and can be answered in a short span of time.