Question

Let $f(x)=2{{x}^{2}},~g(x)=3x+2~and~fog(x)=18{{x}^{2}}+24x+c$, Then c=
A. 2
B. 8
C. 6
D. 4

Answer 1

Hint: In this question, we will use the concept of the composite of functions. fog is a composite function which means g(x) function is in the f(x) function. We have given the expression for f(x) and g(x). Now substitute these expressions in fog(x) and obtain an expression. On comparing you will get the value of the unknown.

Formula Used: The composition of functions is given by: fog(x)= f[g(x)]
Here f(x) and g(x) are two functions.

Complete step by step solution: We have given:
$f(x)=2{{x}^{2}}\\
g(x)=3x+2\\
fog(x)=18{{x}^{2}}+24x+c…(i)$
Since the composition of functions is given by:
$fog(x)=f\{g(x)\}\\
fog(x)=f(3x+2)\\
fog(x)=2{{(3x+2)}^{2}}\\
fog(x)=18{{x}^{2}}+24x+8…(ii)$
On comparing equation (i) with equation (ii) we have
$\therefore c=8$

So, option B is correct.

Note: The process of merging two or more functions into one function is called the composition of functions. Keep in mind that each statement for the composition uses the same letters in the same order.
You should begin with function g since f (g(x)) makes this clear (innermost parentheses are done first).
Also, fog cannot exist unless the range of g is a subset of the domain of f(x). A mathematical function with real values is one that has real numbers as its values.