Unit 9: Combinations of Functions Review

Unit 9:Combinations of FunctionsReview By. Amar Gill & SagarMody P.4 Ms. Rishad

Agenda 9.2– Combining Two Functions: Sums and Differences 9.3 – Combining Two Functions: Products 9.4 -Exploring Quotients of Functions 9.5 - Composition Function

9.2 Combining Two Functions: Sums and Differences Key Ideas • When two functions f(x) and g(x) are combined to form the function (f + g)(x), the new function is called the sum of f and g(f + g)(x) = f(x) + g(x) • Similarly, the difference of two functions, f – g, is (f – g)(x) = f(x) – g(x) • The domain of f + g or f – g is the intersection of the domainsof f and g

9.2 Combining Two Functions: Sums and Differences Example 1: Given f(x) = 3x² + 2 and g(x) = 1/x-4 find: • a) (f + g)(x)= 3x² + 2 + 1/x-4b) (f – g)(x)= 3x² + 2 – 1/x-4c)the domain of (f + g)(x)D=(xeR| x=/ 4)

9.2 Combining Two Functions: Sums and Differences • d) (f + g)(1) • = (f + g)(x) = 3x ² + 2 + 1/x-4= (f + g)(1) = 3(1)² + 2 + 1/1-4 • = 3+2 + 1/-3 = 5 – 1/3 • = 15 – 1 3= 14/3

9.3 – Combining Two Functions: Products Key Ideas • When two functions, f(x) and g(x) are combined to form the function, (f x g)(x), the new function is called the product function of f and g To determine the product function, (f x g)(x) • Multiply the values of f(x) by the corresponding y-values of g(x) • (f x g)(x) = f(x) x g(x) • The Domain of (f x g)(x) is the intersection of the domain of f(x) and g(x) • If f(x) = 0 or g(x) = 0, then (f x g)(x) = 0

9.3 – Combining Two Functions: Products

9.3 – Combining Two Functions: Products Example 1: Given f(x) = √ x-3 and g(x) = 2x +1 • a) Find (f x g)(x)= f(x) x g(x)= √x-3 (2x +1) • b) Determine (f x g)(12)= √ 12-3 (2(12) + 1) • = 3(25) • =75

9.3 – Combining Two Functions: Products c) Graph  d) Domain and RangeD = (xeR|x > = 3)R = (yeR| y > = 0)

9.4 Exploring Quotients of Functions Key Ideas • When two functions, f(x) and g(x) are combined to form the function (f/g)(x) or f(x) / g(x), the new function is called the quotient of f and g. • Algebraically, (f/g)(x) = f(x) / g(x) • If f(x) = 0 when g(x) isn't 0, then (f/g)(x) = 0 • To graph (f/g)(x), use a TOV

9.4 Exploring Quotients of Functions Example 1: Write the equation of (f/g)(x), graph it, and state the domain F(x) = 5 g(x) = x • (f/g)(x) = 5/x • F(x) D= (xeR) • G(x) D =(xeR) • (f/g)(x) D=(xeR|x=/0) * Reminder* HA is y =a/cex:  f(x)= 4x 2x+1  HA = 2 VA = -1/2

9.5 Composition Function Composition Function • A function that is the composite of two other functions; the function f(g(t)) is called the composite of f with g; the function f(g(t)) is denoted by (f o g)(t) and is defined by using the output of the function g as the input for the function f. Key Ideas • Two functions, f and g, can be combined using a process called composition, which can be represented by f(g(t)) • The output for the inner function, g, is used as the input for the outer function • The function f(g(t)) can be denoted by (f o g)(x) • In most cases, (f o g)(x) does not equal (g o f)(x) because the order in which the functions are composed matters

9.5 Composition Function Example 1: If f(x) = 2x+3 and g(x) = x², find the following • A) (f o g)(3) = f(g(3)) = f(9) = 2(9)+3 = 21 b) (f o g)(x) = f(g(x)) = f(x ²) = 2x ²+3 b) (f o f)(x) = f(2x+3) =2(2x+3)+3 = 4x + 6 + 3 = 4x + 9

Unit 9: Combinations of Functions Review