1.7, page 209 Combinations of Functions; Composite Functions Objectives Find the domain of a function. Combine functions using algebra. Form composite functions. Determine domains for composite functions.

Using basic algebraic functions, what limitations are there when working with real numbers?  A) You canNOT divide by zero. Any values that would result in a zero denominator are NOT allowed, therefore the domain of the function (possible x values) would be limited. B) You canNOT take the square root (or any even root) of a negative number. Any values that would result in negatives under an even radical (such as square roots) result in a domain restriction.

Reminder: Domain Restrictions (WRITE IT DOWN!) For FRACTIONS:  No zero in denominator! For EVEN ROOTS:  No negative under radical!

Check Point 1, page 211 Find the domain of each function.

The Algebra of Functions, pg 213

Finding Domain of f(x) plus, minus, times, or divided by g(x)

See Example 2, page 213. Check Point 2, page 214.  Let f(x) = x – 5 and g(x) = x 2 – 1. Find a) (f + g)(x) b) (f - g)(x) c) (fg)(x) d) (f/g)(x)

See Example 3, page 214. Check Point 3

Given two functionsf andg, the composite function, denoted by fg  (read as “f of g of x”) is defined by

Composition of Functions See Example 4, page 217. Check Point 4 Given f(x)=5x+6 and g(x)=2x 2 – x – 1, find a) f(g(x)) b) g(f(x))

The domain of fg  is the set of all numbers x in the domain of g such that g( x) is in the domain off. Domain of fg  Note: Finding the domain of f(g(x)) is NOT the same as finding f(x) + g(x)

How to find the domain of a composite function 1. Find the domain of the function that is being substituted (Input Function) into the other function. 2. Find the domain of the resulting function (Output Function). 3. The domain of the composite function is the intersection of the domains found above.

See Example 5, page 218.  Check Point 5

Extra Domain Example  Find the domain  There are x’s under an even radical AND x’s in the denominator, so we must consider both of these as possible limitations to our domain.

Extra Example: Operations with Functions Given that f(x) = x and g(x) = x + 2, find: a) (f+g)(x) = b) (f-g)(x) = c) (fg)(x) = d) (f/g)(x) =

Remember: f(x) = x and g(x) = x + 2. Now, let’s find the domain of each answer. a) (f+g)(x) = x 2 + x - 2 b) (f-g)(x) = x 2 – x - 6 c) (fg)(x) = x 3 – 2x 2 – 4x - 8 d) (f/g)(x) = x – 2

Extra Example: Composition Given f(x)=2x – 5 & g(x)=x 2 – 3x + 8, find a) (f◦g)(x) and (g◦f)(x) b) (f◦g)(7) and (g◦f)(7) c) What is the domain of these composite functions?