Question

If f (x) = 2x – 1 and g (x) = 3x + 2 then find (fog) (x)
(a). 2 (3x + 1)
(b). 2 (3x + 2)
(c). 3 (2x + 1)
(d). 3 (3x + 1)

Answer 1

Hint: The notation (fog) (x) can also be written as f (g (x)) which means it represents f (x), where x is replaced by function g (x). So, in f (x) instead of x put the function g (x) or 3x + 2 to get the answer.

Complete step-by-step answer:

In the question we are given two functions f (x) and g (x) as (2x - 1) and (3x + 2) respectively and from this we have to find the function (fog) (x).
Now before proceeding we will briefly learn about functions.
In mathematics, a function is a binary relation over two sets that associates to every element of the first set exactly one element of the second set. Typical examples are functions from integers or from real numbers to real numbers.
A function is a process or relation that associates each element x of a set X, the domain of the domain of the function, to a single element y of another set Y (possibly the same set) the co – domain of the function. If the function is called f, then the relation is denoted by y = f (x), the element x is the argument or input of the function and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function.
The notation (fog) (x) can also be written as f (g(x)) which means that f (g(x)) can be attained by substituting g (x) in place of x in f (x).
So, we are given f (x) as 2x – 1 and g (x) as 3x + 2.
Hence to find (fog) (x) we have consider f (x) and in place of x we will substitute g (x) so we get,
f (x) = 2x – 1 then f (g (x)) = f (3x + 2)
Hence, f (3x + 2) = 2 (3x + 2) – 1
f (3x + 2) = 6x + 4 – 1
f (3x + 2) = 6x + 3
So, f (3x + 2) = 3 (2x + 1)
Hence (fog) (x) is 3 (2x + 1).
So the correct option is (c).

Note: A function is uniquely represented by a set of all pairs (x, f (x)) called a graph of the function. When the domain and co – domain are sets of real numbers, each such pair may be considered as Cartesian co – ordinates of a point in a plane. The set of these points is called a graph of the function.