Question

How do you find \[\left( {\dfrac{g}{f}} \right)\left( 3 \right)\] given \[g(a) = 3a + 2\] and \[f(a) = 2a - 4\] ?

Answer 1

Hint: We need to solve the given functions with performing the addition, subtraction, multiplication and division. Here, the required function values are given in the form of polynomial expression. In order to find the function of quotient rule by substitute the two functions.

Complete step by step solution:
To simplify the following quotient function, we have
 \[\left( {\dfrac{g}{f}} \right)\left( 3 \right)\] , where the given function, \[g(a) = 3a + 2\] and \[f(a) = 2a - 4\] .
By rewrite the quotient function, we get
 \[\dfrac{{g\left( 3 \right)}}{{f\left( 3 \right)}} \to \left( 1 \right)\]
Let assume the function of value \[a = 3\] ,
By substitute the value in the two function \[g(a)\] and \[f(a)\] ,
The polynomial expression for the numerator function,
 \[g(a) \Rightarrow g(3) = 3(3) + 2 = 9 + 2\]
 \[g(3) = 11\]
The polynomial expression for the denominator function, we get
 \[f(a) \Rightarrow f(3) = 2(3) - 4 = 6 - 2\]
 \[f(3) = 2\]
To plug the two polynomial expression value to the quotient function,
 \[\dfrac{{g\left( 3 \right)}}{{f\left( 3 \right)}} = \dfrac{{11}}{2}\]
To simplify the function, we get
 \[\dfrac{{g\left( 3 \right)}}{{f\left( 3 \right)}} = 5.5\]
Hence, the polynomial expression \[\left( {\dfrac{g}{f}} \right)\left( 3 \right)\] is \[5.5\]
So, the correct answer is “ \[5.5\] ”.

Note: Here, a function is given inside another function and it is done by substituting one function into another function is called a composite function is used to solve the quotient rule of the given function and the value. First we have to perform the operation of add, subtract, multiply and division is no more complicated than the notation itself.
By using the quotient rule function of algebraic polynomial expression also given in the problem, here we need to perform mathematical operation to find the function value and then perform the quotient rule is applicable. This is the composite function method defined as composition of a function is done by substituting one function into another function.