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.