Question

Find \[\left( gof \right)\left( x \right)\] and \[\left( gof \right)'\left( x \right)\] , if \[f\left( x \right)={{x}^{5}}\] and \[g\left( x \right)=2x-3\]

Answer 1

Hint:In this question, from the given functions of g and f we can get the composition function of the given both functions accordingly from the formula. Then doing derivation of the composition function obtained with respect to x gives the result.
\[\left( gof \right)\left( x \right)=g\left( f\left( x \right) \right)\]


Complete step-by-step answer:
Now, let us look at the function composition
Given two functions \[f:X\to Y\]and \[g:Y\to Z\] such that the domain of g is the codomain of f, their composition is the function \[gof:X\to Z\] defined by
\[\left( gof \right)\left( x \right)=g\left( f\left( x \right) \right)\]
Now, from the given functions in the question we have
\[\begin{align}
  & f\left( x \right)={{x}^{5}} \\
 & g\left( x \right)=2x-3 \\
\end{align}\]
Now, let us assume the function f to be equal to some y
\[\begin{align}
  & \Rightarrow f\left( x \right)=y \\
 & \Rightarrow y={{x}^{5}} \\
\end{align}\]
Now, as already given in the question that the function g is
\[\Rightarrow g\left( x \right)=2x-3\]
Now, to find the composition function we need to find
\[\Rightarrow g\left( f\left( x \right) \right)\]
Now, on substituting the assumed value of the function f we can further write it as
\[\Rightarrow g\left( y \right)\]
Now, on replacing x with y in the given function of g we get,
\[\Rightarrow 2y-3\]
Now, on replacing again y with the value we assumed earlier we get,
\[\Rightarrow 2{{x}^{5}}-3\text{ }\left[ \because y={{x}^{5}} \right]\]
Thus, we get that
\[\therefore \left( gof \right)\left( x \right)=2{{x}^{5}}-3\]
Now, to find the value of \[\left( gof \right)'\left( x \right)\] we need to find the derivation of the above composition function obtained
\[\Rightarrow \left( gof \right)'\left( x \right)\]
Now, on differentiating the composition function with respect to x we get,
\[\Rightarrow \dfrac{d\left( gof \right)\left( x \right)}{dx}\]
Now, on substituting the respective function we get,
\[\Rightarrow \dfrac{d\left( 2{{x}^{5}}-3 \right)}{dx}\]
Now, on further simplification we get,
\[\Rightarrow 2\times 5\times {{x}^{5-1}}\]
Now, on simplifying it further we get,
\[\Rightarrow 10{{x}^{4}}\]
\[\therefore \left( gof \right)'\left( x \right)=10{{x}^{4}}\]
Hence, \[\left( gof \right)'\left( x \right)=10{{x}^{4}}\]

Note:
Instead of assuming some value for function f we can directly find the composition function by directly substituting the function values. Both the methods give the same result.
It is important to note that while substituting the functions to find the composition function we need to substitute the values accordingly because interchanging any of the functions and wrong derivation changes the result completely