Combinations of Functions Section 1.5 Combinations of Functions

Combinations of Functions Arithmetic Combinations of Functions The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. In the Case of f(x)/g(x), there is the further restriction that g(x)  0. Sum, Difference, Product and Quotient Functions 1. Sum 2. Difference 3. Product 4. Quotient

Combinations of Functions Try these: Given Find: 1. (f + g)(x) 3. (f g)(x) 2. (f – g)(x) 4. (f / g)(x)

Combinations of Funcitons Try these: Given: Find: 1. (f + g)(3) = 8 2. (f – g)(-1) = - 8 3. (f g)(-2) = - 35 4. (g / f)(0) = -1/5

Combinations of Functions Definition of Composition of Two Functions The composition of the function f with the function g is The domain of f ◦ g is the set of all x in the domain of g such that g(x) is in the domain of f.

Combinations of Functions How to from the composite of two functions: Given: Step 1: Think of f(x) as being the primary function and g(x) being the secondary function. To form the composite substitute the secondary function into the primary function. Definition of composite Substitution of 4 – x2 for x in x + 2 Step 2: Simplify the result.

Combinations of Functions How to from the composite of two functions (cont.): Given: Step 1: Think of g(x) as being the primary function and f(x) being the secondary function. To form the composite substitute the secondary function into the primary function. Definition of composite Substitution of x + 2 for x in 4 – x2 Step 2: Simplify the result.

Combinations of Functions How to find the numeric value for the combination of two functions Given: Step 1: Find g(2). Step 2: Take the value for g(2) and substitute it into f(x). Step 3: The answer found in step 2 is the numeric value for

Combinations of Functions How to find numeric values of composite graphically. Given: Step 1: Graph both functions on individual graphs. Step 2: Find g(0) g(0) = 1 Step 3: Take the value from Step 2 and substitute It in for x in f(x). Find its value on the graph. f(1) = -2 f(x) = x2 - 3 g(x) = 2x + 1

Combinations of Functions Try these

Combinations of Functions What you should have learned: 1. To add, subtract, multiply, and divide functions To find the composition of one function with another function.