Example 1: Use the graph of the function f given below to find the following. a. Find the domain of f and list your answer in interval notation

Transcription

1 When working with the graph of a function, the inputs (the elements of the domain) are always the values on the horizontal ais (-ais) and the outputs (the elements of the range) are always the values on the vertical ais (y-ais). So when finding the domain of a function based on its graph, we will use the horizontal ais to determine how far left and right the graph goes. And when finding the range of a function based on its graph, we will use the vertical ais to determine how far up and down the graph goes. In the case of the graph given below, the graph of the function f starts at the point (2, 1) and goes up and to the right. The smallest -value (or input) of the function f is 2, and from there the graph moves off to the right without stopping, so goes to infinity. Therefore the domain of the function f is [2, ). The smallest function value (or output) of the function f is 1, and from there the graph goes up without stopping, so f() goes to infinity. Therefore the range of the function f is [ 1, ). f() The smallest input of a function will always be the -coordinate of the point where the graph starts, unless the graph does not have a starting point, as we ll see on Eamples 3 and 5. The smallest output of a function will not necessarily be the y-coordinate of the point where the graph starts, as we ll see on Eample 1. Ll

2 Eample 1: Use the graph of the function f given below to find the following. f() a. Find the domain of f and list your answer in b. Find the range of f and list your answer in c. Find f( 4) d. Find f(0) e. Find f(2) d. A e.

3 f() f. Find all the -values, such that f() = 2 g. Find all the -values, such that f() = 2 h. Find all the -values, such that f() = 0 The -values that make a function equal to zero are called the Zeros of the Function. These are simply the values on the horizontal ais (the -ais) where the graph of the function touches or crosses the ais. The Zeros of a Function are important not only for identifying the -intecepts of a function, but also for determining the intervals where a function is positive or negative. These intervals are formed by the zeros of the function.

4 Eample 2: Use the graph of the function f given below to find the following. f() a. Find the zeros of the function (f() = 0 when =?), and label them on the graph of f(). b. Find all the -values, such that f() > 0 and list your answers in c. Find all the -values, such that f() < 0 and list your answers in d. Find all the intercepts of the function and list your answers as ordered pairs

5 Eample 3: Use the graph of the function h given below to find the following. h() a. Find the domain of h and list your answer in b. Find the range of h and list your answer in c. Find h( 5) d. Find h(0) e. Find h(3) d. e.

6 h() f. Find all the -values, such that h() = 5 g. Find the zeros of the function (h() = 0 when =?) h. Find all the -values, such that h() > 0 and list your answers in i. Find all the -values, such that h() < 0 and list your answers in j. Find all the intercepts of the function and list your answers as ordered pairs

7 Eample 4: Use the graph of the function g given below to find the following. In each case, think about whether you are looking for an input or an output. g() a. Find the domain of g and list your answer in [ 9, 3] b. Find the range of g and list your answer in [ 1,2] c. Find g( 7) g( 7) = 1 d. Find g(0) g(0) = 2 e. Find g(3) g(3) = 1

8 g() f. Find all the -values, such that g() = 1 = 9, 3, 3, because g( 9) = 1, g( 3) = 1, and g(3) = 1 g. Find all the -values, such that g() = 3 There is no -value that will make g() = 3, so NONE. h. Find all the -values, such that g() = 0 and list your answers in = 8, 4, 2, 2, because g( 8) = 0, g( 4) = 0, g( 2) = 0 and g(2) = 0

9 g() i. Find all the -values, such that g() > 0 and list your answers in g() > 0 when is part of the interval ( 8, 4) ( 2, 2) j. Find all the -values, such that g() < 0 and list your answers in g() < 0 when is part of the interval [ 9, 8) ( 4, 2) (2, 3] k. Find all the intercepts of the function and list your answers as ordered pairs intercepts: ( 8, 0), ( 4, 0), ( 2, 0), (2, 0) y intercept: (0, 2)

10 Eample 5: Use the graph of the function j given below to find the following. j() a. Find the domain of j and list your answer in b. Find the range of j and list your answer in c. Find j( 4) d. Find j(0) e. Find j(5)

11 j() f. Find all the -values, such that j() = 5 g. Find the zeros of the function (j() = 0 when =?) h. Find all the -values, such that j() > 0 and list your answers in

12 j() i. Find all the -values, such that j() < 0 and list your answers in j. Find all the intercepts of the function and list your answers as ordered pairs

13 Eample 6: Use the graph of the function k given below to find the following. k() a. Find the domain of k and list your answer in b. Find the range of k and list your answer in c. Find k( 1) d. Find k(0) e. Find k(4)

14 k() f. Find all the -values, such that k() = 3 g. Find the zeros of the function (k() = 0 when =?) h. Find all the -values, such that k() > 0 and list your answers in

15 k() i. Find all the -values, such that k() < 0 and list your answers in j. Find all the intercepts of the function and list your answers as ordered pairs

16 Answers to Eamples: 1a. [ 6, 2] ; 1b. [ 2, 2] ; 1c. f( 4) = 2 ; 1d. f(0) = 2 ; 1e. f(2) = 0 ; 1f. f() = 2 when = 4 ; 1g. f() = 2 when = 0 ; 1h. f() = 0 when = 6, 2, 2 ; 1i. f() > 0 when is part of the interval ( 2, 2) ; 1j. f() < 0 when is part of the interval ( 6, 2) ; 2a. f() = 0 when = 6, 2, 2 ; 2b. ( 2, 2) ; 2c. ( 6, 2) ; 2d. ( 6, 0), ( 2, 0), (2, 0), (0, 2) ; 3a. (, ) ; 3b. (, ) ; 3c. f( 5) = 5 ; 3d. f(0) = 0 ; 3e. 3(3) = 3 ; 3f. f() = 5 when = 5 ; 3g. f() = 0 when = 4, 0, 4 ; 3h. f() > 0 when is part of the interval ( 4, 0) (4, ) ; 3i. f() < 0 when is part of the interval (, 4) (0, 4) ; 3j. ( 4, 0), (0, 0), (4, 0) ; 4a. [ 9, 3] ; 4b. [ 1, 2] ; 4c. g( 7) = 1 ; 4d. g(0) = 2 ; 4e. g(3) = 1 ; 4f. g() = 1 when = 9, 3, 3 ; 4g. g() 3 ; 4h. g() = 0 when = 8, 4, 2, 2; 4i. g() < 0 when is part of the interval [ 9, 8) ( 4, 2) (2, 3] ; 4j. ( 8, 0), ( 4, 0), ( 2, 0) (2, 0) (0, 2) ; 5a. (, ) ; 5b. (, ) ; 5c. j( 4) = 5 ; 5d. j(0) = 3 ; 5e. j(5) = 4 ; 5f. j() = 5 when = 8 ; 5g. j() = 0 when = 6, 3, 1 ; 5h. j() > 0 when is part of the interval ( 6, 3) (1, ) ; 5i. j() < 0 when is part of the interval (, 6) ( 3, 1) ; 5j. ( 6, 0), ( 3, 0), (1, 0), (0, 3) ; 6a. [0, ) ; 6b. [0, ) ; 6c. k( 1) is undefined ; 6d. k(0) = 0 ; 6e. k(4) = 2 ; 6f. k() = 3 when = 9 ; 6g. k() = 0 when = 0 ; 6h. k() > 0 for every value of in its domain (0, ) ; 6i. k() is never less than 0 ; 6j. (0, 0) ;