Question

How do you find $f\left( x \right)$ and $g\left( x \right)$ such that $h\left( x \right) = \left( {f \circ g} \right)\left( x \right)$ and $h\left( x \right) = {\left( {8 - 4x} \right)^2}$

Answer 1

In order to solve this sum, we need to be familiar with composite functions. Here function $g\left( x \right)$ is inside function $f\left( x \right)$, which means that we simply place the values of function $g\left( x \right)$ in $f\left( x \right)$ .Now since, $h\left( x \right) = \left( {f \circ g} \right)\left( x \right)$ and $h\left( x \right) = \left( {8 - 4{x^2}} \right)$, therefore we can easily find the values of functions $f\left( x \right)$ and $g\left( x \right)$ as $g\left( x \right)$ is equal to the numbers on the inside while $f\left( x \right)$ is equal to operation on the outside as whole.

Complete step-by-step solution:
The given conditions are: $h\left( x \right) = \left( {f \circ g} \right)\left( x \right)$ and $h\left( x \right) = {\left( {8 - 4x} \right)^2}$
Here, $\left( {f \circ g} \right)\left( x \right)$ means that function $g\left( x \right)$ is inside function $f\left( x \right)$.
These functions are then said to be composite functions of each other.
Thus, $\left( {f \circ g} \right)\left( x \right)$ can also be represented as $f\left( {g\left( x \right)} \right)$
Now, according to the question: $h\left( x \right) = \left( {f \circ g} \right)\left( x \right)$
Therefore, $f\left( {g\left( x \right)} \right) = h\left( x \right)$ and $h\left( x \right) = {\left( {8 - 4x} \right)^2}$
Thus, $f\left( {g\left( x \right)} \right) = {\left( {8 - 4x} \right)^2}$, now we need to decompose this composite function so that we can find our $f\left( x \right)$ and $g\left( x \right)$.
Decomposing the function means to simply find the inverse of the original $f\left( {g\left( x \right)} \right)$ .
In the above mentioned function, $f\left( x \right)$ simply represents the numbers present on the outside while $g\left( x \right)$ represents the numbers present inside.
Since, $f\left( {g\left( x \right)} \right) = {\left( {8 - 4x} \right)^2}$, therefore :
$g\left( x \right) = 8 - 4x$, Since $g\left( x \right)$ represents the numbers present inside.
$f\left( x \right) = {\left( x \right)^2}$, Since $f\left( x \right)$ represents the function on the outside as a whole.
Hence we get the required answer.

Note: Composite functions or functions composition is simply an operation that takes two functions, $f\left( x \right)$ and $g\left( x \right)$. Then apply the result of one on to the other to give rise to another function, say, $h\left( x \right)$ The composite function is denoted by the symbol ‘$\circ$ ‘ . It is different from the multiplication symbol which is denoted as ‘ ⋅’.
Some common properties of composite functions are:
They are commutative if $g \circ f = f \circ g$
They are associative if $f,g,h$ are compostable then $f \circ \left( {g \circ h} \right) = \left( {f \circ g} \right) \circ h$