Question

How do you find \[\left( fog \right)\left( n \right)\] given \[f\left( n \right)=2n\] and \[g\left( n \right)=-n-4\]?

Answer 1

Hint: This question belongs to the topic of algebra. The function \[\left( fog \right)\left( n \right)\] is a type of composite function. In this question, first we will know about composite function. After that, we will put the value of n as the function \[g\left( n \right)\]in the function \[f\left( n \right)\] to get the value of \[\left( fog \right)\left( n \right)\]. After that, we will get the answer.

Complete step by step answer:
Let us solve this question.
In this question, we have to find the value of \[\left( fog \right)\left( n \right)\] for which we have given as \[f\left( n \right)=2n\] and \[g\left( n \right)=-n-4\]. That means we have to put the value of \[g\left( n \right)\] as n in the function\[f\left( n \right)\].
This type of function that we have to find is \[\left( fog \right)\left( n \right)\] called a composite function. A composition function is a function inside of a function.
We can write the function \[\left( fog \right)\left( n \right)\] as in the form \[f\left( g\left( n \right) \right)\] also.
Now, we will find the value of\[f\left( g\left( n \right) \right)\].
\[\left( fog \right)\left( n \right)\] can also be written as
\[\left( fog \right)\left( n \right)=f\left( g\left( n \right) \right)\]
We can write the value of \[g\left( n \right)\] as \[-n-4\] in the above equation, we get
\[\Rightarrow \left( fog \right)\left( n \right)=f\left( -n-4 \right)\]
Now, as we know that the value of \[f\left( n \right)\] is \[2n\]. So, we can write the value of \[f\left( -n-4 \right)\] by putting \[\left( -n-4 \right)\] in the place of n in\[f\left( n \right)\]. Then, we will get the above equation as
\[\Rightarrow \left( fog \right)\left( n \right)=f\left( -n-4 \right)=2\left( -n-4 \right)\]
The above equation can also be written as
\[\Rightarrow \left( fog \right)\left( n \right)=-2n-8\]

Hence, we get that the value of \[\left( fog \right)\left( n \right)\] is \[-2n-8\].

Note: We should have a better knowledge in the topic of algebra. We should know about functions to solve this type of question easily. And, also don’t forget about the composite functions. Always remember that whenever we have to find the value of composite functions, then we will put the value of the second term in place of input of the first term and solve them accordingly, we will get the answer.