6.2 Function Operations

©2001 by R. Villar

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Function Operations

You can perform operations, such as addition, subtraction, multiplication, and division, with functions…

For example:

If f(x) = 3x and g(x) = x – 5

f(x) + g(x) = 3x + (x – 5)

= 4x – 5

f(x) – g(x) = 3x – (x – 5)

= 2x + 5

f(x) • g(x) = 3x(x – 5)

= 3x2 – 15x

What about f(x) ÷ g(x) ?

f(x) ÷ g(x) = 3x x – 5

Be sure to consider the domain (the possible inputs)…

The domain is does not include 5 since that would make the denominator 0…

Therefore, the domain is all real numbers except x = 5.

Example: Find the domain of

Think of the values that will make the square root a negative number

The domain is all real numbers greater than 1. The graph will confirm this...

f (x) x 1

Composition of Functions

If there is a 40% off sale at Nordstrom’s and as an employee you receive a 10% discount, how much will you pay on a $299 jacket?

You do not get 50% off…

...this is an example of a composite function.

You will pay 90% of the cost (10% discount) after you pay 60% (40% discount).

The two functions look like this…f(x) = 0.6x g(x) = 0.9x

We can put these together in a composite function that looks like this… f(g(x))

“f of g of x”

Work from the inside out (find g of 299 first)...

f(g(299)) = f(0.9 • 299)

= f(269.1)

Now, find f of 269.1...

= 0.6 • 269.1

= 161.46

The jacket will cost $161.46

If f(x) = 0.6x and g(x) = 0.9xWhat is f(g(x)) when x = 299?

Examples:If f(x) = x2 – 5 and g(x) = 3x2 + 1

find f[g(2)] and g[f(2)] f[g(2)] = f[3(2)2 + 1]f[g(2)] = f[13]f[g(2)] = (13)2 – 5f[g(2)] = 164g[f(2)] = g[(2)2 – 5]g[f(2)] = g[ – 1 ]g[f(2)] = 3(–1)2 + 1g[f(2)] = 4

Example: If f(x) = x2 + 6 and g(x) = 3x – 4find f(g(x))

f[g(x)] = f(3x – 4) = (3x – 4)2 + 6 = 9x2 – 24x + 16 + 6 = 9x2 – 24x + 22