Question

Find gof and fog if \[f\left( x \right)=8{{x}^{3}}\] and \[g\left( x \right)={{x}^{\dfrac{1}{3}}}.\]

Answer 1

Hint: To solve the given question, we will first find out what kind of functions are given in the question. Then we will make use of the fact that if A(x) and B(x) are two functions, then we can write (AoB)(x) as A(B(x)). Then, to find out the value of A(B(x)), we will put B(x) in the place of x in A(x). In this way, we will get the functions gof and fog. The identity \[{{a}^{n}}\times {{b}^{n}}={{\left( a\times b \right)}^{n}}\] will be made use of.

Complete step-by-step answer:
Before we start to solve the given question, we must know that the function gof and fog are composite functions. A composite function is a function that depends on another function. A composite function is formed when one function is substituted into another function. The general form of a composite function is (AoB)(x) which is read as “A of B of x”. It is formed when B(x) is substituted for x in A(x). Now, we will make use of the fact that we can write (AoB)(x) as A(B(x)). Thus, we can write gof as g(f(x)). Now, we have to find the function g(f(x)). For this, we will replace x with f(x) in the function g(x). Thus, we will get,
\[g\left( f\left( x \right) \right)={{\left( f\left( x \right) \right)}^{\dfrac{1}{3}}}\]
\[\Rightarrow g\left( f\left( x \right) \right)={{\left( 8{{x}^{3}} \right)}^{\dfrac{1}{3}}}\]
\[\Rightarrow g\left( f\left( x \right) \right)={{\left[ {{\left( 2 \right)}^{3}}{{\left( x \right)}^{3}} \right]}^{\dfrac{1}{3}}}\]
Now, we will use the identity \[{{a}^{n}}\times {{b}^{n}}={{\left( a\times b \right)}^{n}}\]
\[\Rightarrow g\left( f\left( x \right) \right)={{\left[ {{\left( 2x \right)}^{3}} \right]}^{\dfrac{1}{3}}}\]
By using the identity \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{m\times n}}\] we get,
\[\Rightarrow g\left( f\left( x \right) \right)={{\left( 2x \right)}^{3\times }}^{\dfrac{1}{3}}\]
\[\Rightarrow g\left( f\left( x \right) \right)={{\left( 2x \right)}^{1}}\]
\[\Rightarrow g\left( f\left( x \right) \right)=2x\]
\[\Rightarrow \left( gof \right)\left( x \right)=2x\]
Similarly, we can write fog as f(g(x)). Now, to find the function f(g(x)), we will replace x with g(x) in f(x). Thus, we will get,
\[f\left( g\left( x \right) \right)=8\left[ {{\left( g\left( x \right) \right)}^{3}} \right]\]
\[\Rightarrow f\left( g\left( x \right) \right)=8\left[ {{\left( {{x}^{\dfrac{1}{3}}} \right)}^{3}} \right]\]
\[\Rightarrow f\left( g\left( x \right) \right)=8\times {{x}^{\dfrac{1}{3}\times 3}}\]
\[\Rightarrow f\left( g\left( x \right) \right)=8x\]
\[\Rightarrow \left( fog \right)\left( x \right)=8x\]

Note: As the domains and ranges of both the functions given in the question is R, i.e. \[\left( -\infty ,\infty \right),\] we do not need to check whether fog and gof exist or not. But if the functions of which we have to find the composite function are restricted to the interval, then we need to check whether the range of the function (inner function) is equal to the domain of the outer function or not.