Use the table to evaluate each expression.

x | 1 2 3 4 5 6
f | 3 1 4 2 2 5
g | 6 3 2 1 2 3

a). f(g(1))
b). g(f(1))
c). f(f(1))
d). g(g(1))
e). (gof)(3)
f). (fog)(6)

explain the concept and show step-by-step solution.

This question uses the concept of f(g(x)) in which the value of a given function is substituted into a second function.
a) f(g(1))
First we evaluate g(1) using the table
g(1) = 6
Now we plug the value of g(1) into the original equation f(g(1))
= f(6); using the table:
= 5
We apply the same method to the remaining ones:
b). g(f(1))
f(1) = 3
g(3) = 3

c). f(f(1))
f(1) = 3
f(3) = 4

d). g(g(1))
g(1) = 6
g(6) = 3

e). (gof)(3)
= g(f(3))
= f(3) = 4
g(4) = 1

f). (fog)(6)
=f(g(6))
g(6) = 3
f(3) = 4