Question

Given \[f\left( 2 \right) = 3,{\text{ }}f\left( 3 \right) = 4,{\text{ }}f\left( 5 \right) = 0,{\text{ }}g\left( 2 \right) = 5,{\text{ }}g\left( 3 \right) = 2,{\text{ }}g\left( 4 \right) = - 1,\] how do you evaluate \[f\left( {g\left( 2 \right)} \right)?\]

Answer 1

Hint: In the given question, you have given a set of values for a set of arguments for two different functions, and you have to calculate the given function whose argument is another function. So first find the value of the argument function then put that value in the parent function and then find its value accordingly.

Complete step by step answer:
In order to evaluate the given functional value, that is \[f\left( {g\left( 2 \right)} \right)\],
Here this type of function is also called nested function, in which a function is inside the another function as we can see that the function $g(x)$ is inside the function $f(x)$ so if we want to evaluate the function \[f\left( {g\left( 2 \right)} \right)\] we have to first evaluate its argument, that is \[g\left( 2 \right)\]
We have following given values of the function $g(x)$
\[g\left( 2 \right) = 5,{\text{ }}g\left( 3 \right) = 2,{\text{ }}g\left( 4 \right) = - 1\]
We want the value of \[g\left( 2 \right) = 5\]
So putting this value in the parent function, we will get
\[f\left( {g\left( 2 \right)} \right) = f\left( 5 \right)\]
Now, again seeing the set of given values for the function $f(x)$ we have
\[f\left( 2 \right) = 3,{\text{ }}f\left( 3 \right) = 4,{\text{ }}f\left( 5 \right) = 0\]
And we need \[f\left( 5 \right) = 0\]

Therefore the function \[f\left( {g\left( 2 \right)} \right) = 0\]

Note: In this question, set of arguments and their values are directly given. But in some questions, you will have the function expression for both the function, so what you have to do is find the value of the argument function first and then replace the independent variable of the parent function with that value to get the required value. You will face functions in which there are more than two functions being nested, so solve them step by step by starting from the very argument which is at last in the nested function.