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Question

If f (x) = x - 1, g(x) = x + 1, and N = {1, 2, 3, 4,. . .} is the domain of each function, find the simplest possible expression for the following.

(a) f (g(x))

(b) g(f(x))

Answer 1

If f (x) = x - 1, g(x) = x + 1, and N = {1, 2, 3, 4,. . .}
So, domain would be x ≥ 1

(a) f (g(x))
f(x) = x - 1
f(g(x)) = (x+1) - 1
f(g(x)) = x, where x ≥ 1

(b) g(f(x))
g(x) = x + 1
g(f(x)) = (x-1) + 1
g(f(x)) = x, where x ≥ 1

Answer 2

this is what is commonly referred to as substitution. I guess this would be the formal way to describe it.
it might be a little clearer if they had used different variables.

f(g(x)) means substitute the function g(x) in place of the variable x in f(x). Similarly
g(f(x)) means substitute the function f(x) in place of the variable x in g(x).

so if f(x) = x-1, and g(x) = x+1

f(g(x)) = g(x)-1 = (x+1)-1 = x+1-1 = x+(1-1) = x+0 = x

g(f(x)) = f(x)+1 = (x-1)+1 = x-1+1 = x+(-1+1) = x+0 = x

Answer 3

f ( g(x) ) = f (x + 1) = x + 1 - 1 = x

g ( f(x) ) = g (x - 1) = x - 1 + 1 = x
-

Answer 4

f(g(x))=f(x + 1)=(x + 1) - 1=x + 1 - 1=x + (1-1)=x + 0 = x
g(f(x))=g(x - 1)=(x - 1) + 1 = ... (you finish it) ... = x

x

for both questions

Answer 5

A