Question

If we have a function as $f\left( x \right) = 2x + 4$, then what is the value of $f\left( 4 \right) + f\left( 6 \right)$?
(A) $f\left( 8 \right)$
(B) $f\left( {10} \right)$
(C) $f\left( {12} \right)$
(D) $f\left( {18} \right)$
(E) $f\left( {28} \right)$

Answer 1

Hint: We are required to find the values of a function for some specific value of the variable and then find the value of the expression given to us. This question requires us to have the knowledge of basic and simple algebraic rules and operations such as substitution, addition, multiplication, subtraction and many more like these. A thorough understanding of functions and its applications can be of great significance.

Complete step-by-step solution:
In the given question, we are required to find the value of the expression $f\left( 4 \right) + f\left( 6 \right)$.
Also, we have the function $f\left( x \right) = 2x + 4$. So, we substitute the value of the variables in the function to find the value of expression. The value of a function at a certain value of variable is found by substituting the value of variable as specified in the question into the function.
So, we need to replace the variable in the function given to us in the question by the value specified.
So, the function given to us is: $f\left( x \right) = 2x + 4$.
We are required to find the value of $f\left( 4 \right)$ and $f\left( 6 \right)$ by replacing the value of variable x in the function by $4$ and $6$ respectively.
Hence, $f\left( 4 \right) = 2\left( 4 \right) + 4$
Multiplying the terms, we get,
$ \Rightarrow f\left( 4 \right) = 8 + 4$
Adding up the terms,
$ \Rightarrow f\left( 4 \right) = 12$
Now, $f\left( 6 \right) = 2\left( 6 \right) + 4$
Multiplying the terms, we get,
$ \Rightarrow f\left( 6 \right) = 12 + 4$
Adding up the terms,
$ \Rightarrow f\left( 6 \right) = 16$
Hence, we get the values of $f\left( 4 \right)$ and $f\left( 6 \right)$ as $12$ and $16$. So, we substitute these values in the expression.
$f\left( 4 \right) + f\left( 6 \right) = 12 + 16$
$ \Rightarrow f\left( 4 \right) + f\left( 6 \right) = 28$
But we have to match the options given in the question. So, we have to calculate the values of options as well.
So, $f\left( 8 \right) = 2\left( 8 \right) + 4 = 16 + 4 = 20$
Similarly, $f\left( {10} \right) = 2\left( {10} \right) + 4 = 24$
$f\left( {12} \right) = 2\left( {12} \right) + 4 = 24 + 4 = 28$
$f\left( {18} \right) = 2\left( {18} \right) + 4 = 36 + 4 = 40$
$f\left( {28} \right) = 2\left( {28} \right) + 4 = 56 + 4 = 60$
So, the value of $f\left( 4 \right) + f\left( 6 \right)$ is the same as $f\left( {12} \right)$ as both are equal to $28$. So, we get, $f\left( 4 \right) + f\left( 6 \right) = f\left( {12} \right)$.
Therefore, the correct answer is option (C).

Note: Such questions that require just a simple change of variable can be solved easily by keeping in mind the algebraic rules such as substitution and transposition. Substitution of a variable involves putting a certain value in place of the variable. That specified value may be a certain number or even any other variable.