1 of 24 Pre-Cal Chapter 1 Sections 5 & 6 SAT/ACT Warm - up

Chapter 1 Sec 5 Combination Functions

3 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Chapter 1.5 – Essential Question How do you combine two functions to form a new function? Key Vocabulary: None

4 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Operations on Functions Given f(x) = 3x 2 – 4 and g(x) = 4x + 5, Find each function. a. (f + g)(x) = f(x) + g(x) = 3x 2 – 4 + 4x + 5 = 3x 2 + 4x + 1 b. (f - g)(x) = f(x) - g(x) = 3x 2 – 4 – (4x + 5) = 3x 2 – 4 – 4x – 5 = 3x 2 – 4x – 9 = (3x 2 – 4)(4x + 5) = 12x x 2 – 16x – 20

5 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Composition of Functions Function can also be combined by using composition. In composition, a function is performed, and then a second function is performed on the result of the first function. Think in term of manufacturing, a fiber is made into cloth, then cloth is made into garment.

6 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Composition of Functions In composition, a function g maps the elements in set R to those in set S Another function f maps the elements in set S to those in set T. So the range of g becomes the domain of f. The function formed by composing two functions f and g is called the composite of f and g, denote by read as “f composition g” or “f of g”

7 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Composite Functions = f(5x – 6) = 2(5x – 6) 2 – 3(5x – 6) + 8 = 2(25x 2 – 60x + 36) – 15x = 50x 2 – 135x + 98 = g(2x 2 – 3x + 8) = 5(2x 2 – 3x + 8) – 6 = 10x 2 – 15x + 40 – 6 = 10x 2 – 15x + 34 Define Composition of Two Functions The composition of the function f with the function g is: (f ◦ g)(x) = f(g(x)), 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. Find (f ◦ g)(x) and (g ◦ f)(x) for f (x) = 2x 2 – 3x + 8 and g(x) = 5x – 6. (f ◦ g)(x) = f(g(x))(g ◦ f)(x) = g(f(x))

8 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Composite Functions What do we know: 1.The domain of f seems to be all real numbers. 2. But the domain of g Remember no negative under radicals is [–3, 3] 3. So, the domain of (f ◦ g) is [–3, 3]. Find the domain of the composition (f ◦ g)(x) for the function given by f(x) = x 2 – 9 and g(x) =. (f ◦ g)(x) = f(g(x))

9 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Identify a Composite Function Solution Write the function h(x) = (3x – 5) 3 as a composition of two functions. One way to write h as a composition of two functions is to take the inner function to be g(x) = 3x – 5 and the outer function to be f(x) = x 3. Then you can write h(x) = (3x – 5) 3 = f(3x – 5) = f(g(x)).

10 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Identify a Composite Function Solution Write the function as a composition of two functions. One way to write h as a composition of two functions is to take the inner function to be g(x) = x – 2 and the outer function to be Then you can write:

11 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Chapter 1.5 – Essential Question How do you combine two functions to form a new function?

12 of 24 Pre-Cal Chapter 1 Sections 5 & 6 Daily Assignment Chapter 1 Sections 5 Text Book (TB) Textbook Pgs 58 – 59 #5, 7, 11, 15, 17, 21, 35, 36, 39, 44, 61, 63; Read Sections 1.6 Show all work for credit.