Question

Let $f:\{2,3,4,5\}\to \{3,4,5,9\} and g:\{3,4,5,9\}\to\{7,11,15\}$ be functions defined as
f(2) =3f(3)=4, f(4)=f(5)=5, g(3)=g(4)=7, and g(5)=g(9)=11. Then gof(5) is equal to
A. 5
B. 7
C. 11
D. 1

Answer 1

Hint:In this question, we will use the concept of the composite of functions. The gof is a composite function which means f(x) the function is in the g(x) function. To determine the value of a composite function g(f(5)) or gof(5), we will use f(5). Substitute the value given at f(5) in gof(5).

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

Complete step by step solution: We have given functions $f:\{2,3,4,5\} \rightarrow\{3,4,5,9\} and g:\{3,4,5,9\} \rightarrow\{7,11,15\}$
At f(2)=3 and g(3)=7 we get;
gof(2)=g(f) 2))=g(3)=7
At f(3)=4 and g(4)=7 we get;
gof(3)=g(f) 3))=g(4)=7
At f(4)=5 and g(5)=11 we get;
gof(4)=g(f(4))=g(5)=11
At f(5)=5 and g(5)=5 we get;
gof(5)=g(f(5))=g(5)=11
Therefore the value of gof(5) equals 11.

Thus, Option (C) is correct.

Note:The composition of functions is the process of combining two or more functions into a single function. Keep in mind that the same letters are used in the same order in each statement of the composition.
Considering that g(f(x)) clearly states this, you should start with the function f (innermost parentheses are done first).