More Functions

Sometimes, functions are written as other functions combined through addition, subtraction, multiplication, or division.  These properties exist for functions with the domain of the intersection of the original functions' domains.

(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f * g)(x) = f(x) * g(x)
(f / g)(x) = f(x) / g(x)    g(x) ≠ 0

For example, find (f + g)(x) if f(x) = 4x - 2 and g(x) = x² + x - 12.

(f + g)(x) = f(x) + g(x) =
(2x - 8) + (x² - x - 12) =
x² + x - 20

For those same functions, find (f / g)(x).

(f / g)(x) = f(x) / g(x) =
(2x - 8) / (x² - x - 12) =
2(x - 4) / (x - 4)(x + 3) =
2 / (x-3)

However, we must put restrictions (values of the variable that will make the denominator = 0) on the domain of this new function, because we have variables in the denominator.  We must remember to make them from the original denominator as well, not after it has been factored.

x - 4 = 0 or x - 3 = 0, x = 4 or 3

So the domain of (f / g)(x) is D = {x | x is a real number other than 3 or 4}.

Another operation performed on functions is composition.  The composition of functions is defined by:

(f ◦ g)(x) = f(g(x))

D = {x | x is in the domain of g so that g(x) is in the domain of f}

Here is an example:
What is (f ◦ g)(x) if f(x) = 4x + 1 and g(x) = x²?

(f ◦ g)(x) = f(g(x)) =
f(x²) = 4(x²) + 1 = 4x² + 1

Try this:
Find (f ◦ g)(x) if f(x) = x³ and g(x) = x + 2.

(f ◦ g)(x) = f(g(x)) =
f(x + 2) = (x + 2)³ =
x³ + 6x² + 12x + 8

Try this:
Find (f ◦ g)(x) if f(x) = √x and g(x) = -x.

(f ◦ g)(x) = f(g(x)) =
f(-x) = √-x

The domain for this new function must be D = {x | x ≤ 0} because there are no square roots of negative numbers, and the radicand, -x, would be negative if x > 0.

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