1 / 19

Warm-up

Warm-up. Sketch the graph of the function. Label the endpoints !. Answers. Use Calculator to check!. Precalculus L esson 1.8. Check: p. A36 #120, 122, 124, 134,140, 152, 156,160, 174, 184. 1.8. Objective: Combine functions arithmetically and graphically. What You Should Learn.

gurit
Download Presentation

Warm-up

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Warm-up Sketch the graph of the function. Label the endpoints!

  2. Answers Use Calculator to check!

  3. PrecalculusLesson 1.8 Check: p. A36 #120, 122, 124, 134,140, 152, 156,160, 174, 184

  4. 1.8 Objective: Combine functions arithmetically and graphically.

  5. What You Should Learn • Add, subtract, multiply, and divide functions. • Find the composition of one function with another function. • Use combinations and compositions of functions to model and solve real-life problems.

  6. Arithmetic Combinations of Functions

  7. Example 1 Given f (x)= 2x – 3 and g(x) = x2 – 1 find the sum, difference, product, and quotient of f and g. f(x) + g(x) = (2x – 3) + (x2 – 1) = x2 + 2x – 4 f(x) – g(x) = (2x – 3) – (x2 – 1) = –x2 + 2x – 2 f(x)g(x) = (2x – 3)(x2 – 1) = 2x3 – 3x2– 2x + 3 Sum Difference Product Quotient

  8. 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 the quotient f(x)/g(x), there is the further restriction that g(x)  0.

  9. Example 2 – Finding the Sum of Two Functions & evaluating. Given f(x) = 2x + 1 and g(x) = x2 + 2x – 1 find (f + g)(x). Then evaluate the sum when x = 3. Solution: (f + g)(x) = f(x) + g(x) = (2x + 1) + (x2 + 2x – 1) When x = 3, the value of this sum is (f + g)(3) = 32 + 4(3) = x2 + 4x = 21. Do Example 1 on notes 1.8

  10. Composition of Functions Figure 1.90

  11. Example 3 – Composition of Functions Given f(x) = x + 2 and g(x) = 4 – x2, find the following. a. (f g)(x) b. (g f)(x) c. (g f)(–2) Solution: a. The composition of f with g is as follows. (f g)(x) = f(g(x)) = f(4– x2) = (4– x2) + 2 = –x2 + 6 Definition of f g Definition of g(x) Definition of f(x) Simplify.

  12. Example 3 – Solution cont’d b. The composition of g with f is as follows. (g f)(x) = g(f(x)) = g(x + 2) = 4 – (x + 2)2 = 4 – (x2 + 4x + 4) = –x2 – 4x Note that, in this case, (f g)(x)  (g f)(x). Definition of g f Definition of f(x) Definition of g(x) Expand. Simplify.

  13. Example 3 – Solution cont’d c. Using the result of part (b), you can write the following. (g f )(–2) = –(–2)2 – 4(–2) = –4 + 8 = 4 Substitute. Simplify. Simplify. Do Example 2 on notes 1.8

  14. Composition of Functions In calculus, it is also important to be able to identify two functions that make up a given composite function.

  15. Composition of Functions Basically, to “decompose” a composite function, h(x) = (3x – 5)3 look for an “inner” function and an “outer” function. g(x) = 3x – 5 is the inner function and f(x) = x3 is the outer function. Therefore, h(x) = f(g(x)) = (3x – 5)3.

  16. Example 4 – decompose the composite functions so that h(x) = f(g(x)) cont’d Do Example 3 on notes 1.8

  17. p. 88 # 19-27 odd, 49, 51 Classwork

  18. Homework Homework(1.8) p. 88 #12,16, 24,28,38, 42,46, 50, 52, 58

  19. Find the domain of f(g(x)) Closure

More Related