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Question: How do you write \(j\left( x \right)={{\sin }^{2}}\left( x \right)\) as a composition of two or more...

How do you write j(x)=sin⁡2(x)j\left( x \right)={{\sin }^{2}}\left( x \right) as a composition of two or more functions ?

Explanation

Solution

A composite function is usually composed of other functions such that the output of one function is the input of the other function. In other words, when the value of a function is found from two other given functions by applying one function to an independent variable and the other to the results of the other function whose domain consists of those values of the independent variable for which the results yielded by the first function lies in the domain of the second. We should check domains and ranges while applying composite functions.

Complete step-by-step solution:
In simple words, the output of one function must be the input of the other functions. So now we have to write our functions as a composition of two or more functions.
We see that our function is squared. So for this, we can use a separate function. And then treat sin⁡x\sin x as one functions.
Let us assume that g(x)=x2g\left( x \right)={{x}^{2}} and another function namely h(x)=sin⁡xh\left( x \right)=\sin x .
Now let us see how we mathematically write the composition of these two functions.
⇒j(x)=goh(x)=g(h(x))\Rightarrow j\left( x \right)=goh\left( x \right)=g\left( h\left( x \right) \right)
We read goh(x)goh\left( x \right) as ggcircle hh or gg circle h(x)h\left( x \right) .
We assumed h(x)=sin⁡xh\left( x \right)=\sin x. Let us substitute it.
Upon doing so, we get the following :
⇒j(x)=goh(x)=g(h(x)) ⇒j(x)=goh(x)=g(sin⁡x) \begin{aligned} & \Rightarrow j\left( x \right)=goh\left( x \right)=g\left( h\left( x \right) \right) \\\ & \Rightarrow j\left( x \right)=goh\left( x \right)=g\left( \sin x \right) \\\ \end{aligned}
We assumed g(x)=x2g\left( x \right)={{x}^{2}}. So sin⁡x\sin x becomes the input for g(x)g\left( x \right) . We substitute sin⁡x\sin x in the place of xx.
Upon doing so, we get the following :
⇒j(x)=goh(x)=g(h(x)) ⇒j(x)=goh(x)=g(sin⁡x) ⇒j(x)=goh(x)=sin⁡2x \begin{aligned} & \Rightarrow j\left( x \right)=goh\left( x \right)=g\left( h\left( x \right) \right) \\\ & \Rightarrow j\left( x \right)=goh\left( x \right)=g\left( \sin x \right) \\\ & \Rightarrow j\left( x \right)=goh\left( x \right)={{\sin }^{2}}x \\\ \end{aligned}
∴\therefore We can write j(x)=sin⁡2(x)j\left( x \right)={{\sin }^{2}}\left( x \right) as a composition of two or more functions by assuming two functions namely h(x)=sin⁡xh\left( x \right)=\sin x,g(x)=x2g\left( x \right)={{x}^{2}}.

Note: We should be very careful with the concept of composition functions. It can be twisted and asked. There are also inverse functions which can be quite tricky. There will also be inverse of composite functions. These two concepts can be given as one question. Huge amount of practice is needed to get clarity with these and do the question quickly in the exam. We should be careful while solving as there is a huge scope of calculation errors.