Properties of functions?

Question

write a paragraph in which you identify a property that holds for the addition of two real numbers but not for the composition of two functions. include examples to support your answer.

Answer 1

The distributive property of the real numbers:

a(b + c) = ab + ac

But the distributive property doesn't hold for the composition of functions:

f(g(x) + h(x)) ≠ f(g(x) + g(h(x))

Here is an example:

f(x) = x²

g(x) = 1

h(x) = 1

Therefore, we have that:

f(g(x) + h(x)) = (1 + 1)² = 2² = 4

But we have that:

f(g(x)) + f(g(x)) = f(1) + f(1) = (1)² + 1² = 2

Clearly, f(g(x) + h(x)) ≠ f(g(x) + g(h(x)).

Hope this helps!

Answer 2

Commutativity:

with real numbers: x + y = y + x.

The order of the addition is not important.

with functions:  f(g(x) is not = g(f(x)), in general

Example:

f(x) = x + 100

g(x) = x ^2

f(g(x)) = x^2 + 100 but

g(f(x)) = (x + 100)^2 = x^2 + 200 * x + 10000

The order of the composition is important.