Question

Let $f(x) = 4x - 3.$ If $f(a) = 9$ and $f(b) = 5$, then calculate $f(a + b)$ .

Answer 1

Hint: In order to solve the given question , we should know the three important concepts related to the question that what is $f(x)$. $f(x)$ is a function , which can be called as “f of x ”, it is a classic way of writing a function . “f ” is the function name and “x” is the input which is put inside the parentheses, that goes into the function which generates an output related somehow to the input . For example - $f(x) = {x^2}$is the function ,
 we give an input of $x = 2$
then the output will be $
   \Rightarrow f(x) = {x^2} \\
   \Rightarrow f(2) = {2^2} \\
   \Rightarrow f(2) = 4 \\
 $
which is the correct answer. The same concept we are going to apply in our given question.

Complete step-by-step solution:
To solve this question we need to approach step by step . The following information is given to us –
Function given , $f(x) = 4x - 3.$
If $f(a) = 9$and $f(b) = 5$then we have to calculate $f(a + b)$.
We will substitute the values of $f(a)$ and $f(b)$ into the function so that we can get values of ‘a’ and ‘b’ .
Putting $f(a) = 9$in Function $f(x) = 4x - 3.$
\[
   \Rightarrow f(x) = 4x - 3 \\
   \Rightarrow f(a) = 4a - 3 \\
 \]
Since , $f(a) = 9$, we get –
\[
   \Rightarrow 4a - 3 = 9 \\
   \Rightarrow 4a = 12 \\
   \Rightarrow a = 3 \\
 \]
Similarly ,
Putting $f(b) = 5$ in Function $f(x) = 4x - 3.$
\[
   \Rightarrow f(x) = 4x - 3 \\
   \Rightarrow f(b) = 4b - 3 \\
 \]
Since , $f(b) = 5$, we get –
\[
   \Rightarrow 4b - 3 = 5 \\
   \Rightarrow 4b = 8 \\
   \Rightarrow b = 2 \\
 \]
Now to calculate $f(a + b)$ substitute the values of ‘a’ and ‘b’ .
$
   \Rightarrow f(a + b) \\
   \Rightarrow f(3 + 2) \\
   \Rightarrow f(5) \\
 $
Putting $ \Rightarrow f(a + b) = f(5)$in function $f(x) = 4x - 3.$, we get-
\[
   \Rightarrow f(x) = 4x - 3 \\
   \Rightarrow f(5) = 4 \times 5 - 3 \\
   \Rightarrow f(5) = 20 - 3 \\
   \Rightarrow f(5) = 17 \\
 \]

Therefore , $f(a + b) = 17$is the required solution .

Note: To perform calculations and to simplify the given function we used the concept of equivalent equations. Equivalent equations are said to be algebraic equations that may have the same solutions if we add or subtract the same number to both sides of an equation - The left-hand side or Right-hand side of the equal sign. Or we can multiply or divide the same number to both sides of an equation - The left-hand side or Right-hand side of the equal to sign with the method of simplification.