Upload
shadow
View
171
Download
14
Embed Size (px)
DESCRIPTION
Section 3.6 Reciprocal Functions. Objectives: 1.To identify vertical asymptotes, domains, and ranges of reciprocal functions. 2.To graph reciprocal functions. Definition. Reciprocal Function Any function that is a reciprocal of another function. 1. sec. x. =. cos x. 1. csc. - PowerPoint PPT Presentation
Citation preview
Section 3.6Reciprocal Functions
Objectives:1. To identify vertical asymptotes,
domains, and ranges of reciprocal functions.
2. To graph reciprocal functions.
Reciprocal Function Any function that is a reciprocal of another function.
DefinitionDefinition
Reciprocal trigonometric ratios:
tan x1
xcot =
sin x1
xcsc =cos x
1xsec =
DefinitionDefinition
Reciprocal trigonometric functions:
y = sec
y = csc
y = cot
DefinitionDefinition
These functions are examples of a larger class of reciprocal functions, including reciprocals of power, polynomial, and exponential functions.
Examples of reciprocal functions
g(x) = 1x2 – 4f(x) = 1
3x4
h(x) = 14x k(x) = sec x
EXAMPLE 1 Find f(1), g(2), h(1/2), and k(/4), using the functions above.
f(x) = 13x4
f(1) = = 13(1)4
13
EXAMPLE 1 Find f(1), g(2), h(1/2), and k(/4), using the functions above.
g(x) = 1x2 – 4
g(2) = = , which is undefined122 – 4
10
EXAMPLE 1 Find f(1), g(2), h(1/2), and k(/4), using the functions above.
h(x) = 14x
h(1/2) = = = 141/2
1 4
12
EXAMPLE 1 Find f(1), g(2), h(1/2), and k(/4), using the functions above.
k(x) = sec x
k(/4) = sec /4 = =1cos /4
1 2/2
2 2
2= =
Since reciprocal functions have denominators, you must be careful about what values are used in the domain.
EXAMPLE 2 Find the domains of f(x), g(x), h(x), and k(x) in the previous functions.Find all values for which the denominator of f(x) and g(x) equals zero.
f(x)3x4 = 0x4 = 0x = 0
g(x)x2 – 4 = 0
x2 = 4x = ±2
EXAMPLE 2 Find the domains of f(x), g(x), h(x), and k(x) in the previous functions.Exclude those values from the domain.
f(x): D = {x|x R, x ≠ 0}g(x): D = {x|x R, x ≠ ±2}
EXAMPLE 2 Find the domains of f(x), g(x), h(x), and k(x) in the previous functions.Since 4x ≠ 0 x, the domain of h(x) is R.Since cos x = 0 when x = /2 + k, k R,the domain of k(x) is D = {x|x R, x ≠ /2 + k, k Z}.
EXAMPLE 3 Graph g(x) = . 1x2 – 4
Give the domain and range. Is g(x) continuous? Is g(x) an odd or even function?
Use reciprocal principles to graph g(x).
EXAMPLE 3 Graph g(x) = . 1x2 – 4
Use reciprocal principles to graph g(x).
EXAMPLE 3 Graph g(x) = . 1x2 – 4
D = {x|x ≠ ±2}
R = {y|y 0 or y -1/4}
g(x) is an even function but is not continuous.
EXAMPLE 3 Graph g(x) = . 1x2 – 4
again without graphing its reciprocal function first.1. Find the domain excluding values
where the denominator equals zero.x2 – 4 = 0x2 = 4x = ±2D = {x|x ≠ ±2}
EXAMPLE 4 Graph g(x) = . 1x2 – 4
again without graphing its reciprocal function first.2. Check for x-intercepts. Since the numerator cannot equal zero, the graph cannot touch the x-axis.
EXAMPLE 4 Graph g(x) = . 1x2 – 4
again without graphing its reciprocal function first.3. Plot a point in each of the regions
determined by the asymptotes (2 & -2). Since the graph cannot cross the x-axis, points within a region will all be on the same side of the x-axis. Include the y-intercept as one of the points.
EXAMPLE 4 Graph g(x) = . 1x2 – 4
again without graphing its reciprocal function first.4. Use the asymptotes as guides. Your
graph will never quite reach either vertical asymptote or the x-axis.
EXAMPLE 4 Graph g(x) = . 1x2 – 4
EXAMPLE 4 Graph g(x) = . 1x2 – 4
EXAMPLE 5 Graph y = csc x. Give the domain, range, zeros, and period. Is it continuous?
EXAMPLE 5 Graph y = csc x. Give the domain, range, zeros, and period. Is it continuous?D = {x|x ≠ k, k Z}R = {y|y 1 or y -1}The function has no zeros; the period is 2. It is not continuous.
Homework:pp. 148-151
►A. Exercises1. f(x)
►A. Exercises3. p(x)
►A. Exercises 6. Give the vertical asymptotes of
h(x) = 1x2 + 5x – 14
►A. ExercisesEvaluate each function as indicated.
9. f(x) = for x = 2 and x = -6 1x2 – 25
►B. Exercises12. Graph the reciprocal function. Give the domain and range.
h(x) = 1x2 + 5x – 14
■ Cumulative Review41. Solve ABC where A = 58°, B = 39°,
and a = 10.5.
■ Cumulative Review42. Give the period and amplitude of
y = 5 sin 3x.
■ Cumulative Review43. Find f(4) if
f(x) =
x – 8 if x 3x2 – 1 if 3 x 97x if x 9
■ Cumulative Review44. How many zeros does a cubic
polynomial function have? Why?
■ Cumulative Review45. Graph y = 2x and estimate 20.7 from the
graph.
A summary of principles for graphing reciprocal functions follows:1. The larger the number, the closer
the reciprocal is to zero.2. The reciprocal of 1 and -1 is itself.3. There is a vertical asymptote for
the reciprocal when f(x) = 0.