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.
Definition
Reciprocal trigonometric ratios:
1
sec x =
cos x
1
cot x =
tan x
1
csc x =
sin x
Definition
Reciprocal trigonometric functions:
y = sec 
y = csc 
y = cot 
These functions are examples
of a larger class of reciprocal
functions, including
reciprocals of power,
polynomial, and exponential
functions.
Examples of reciprocal functions
1
f(x) = 4
3x
1
g(x) = 2
x –4
1
h(x) = x
4
k(x) = sec x
EXAMPLE 1 Find f(1), g(2), h(1/2),
and k(/4), using the functions above.
1
f(x) = 4
3x
1
1
f(1) =
=
4
3(1) 3
EXAMPLE 1 Find f(1), g(2), h(1/2),
and k(/4), using the functions above.
1
g(x) = 2
x –4
1
1
g(2) = 2
= , which is undefined
2 –4 0
EXAMPLE 1 Find f(1), g(2), h(1/2),
and k(/4), using the functions above.
1
h(x) = x
4
1
1 1
1
h( /2) = 1/2 =
=
4
4 2
EXAMPLE 1 Find f(1), g(2), h(1/2),
and k(/4), using the functions above.
k(x) = sec x
1
k(/4) = sec /4 = cos / =
4
2
= = 2
2
1
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)
g(x)
3x4 = 0
x2 – 4 = 0
x4 = 0
x2 = 4
x=0
x = ±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}.
1
EXAMPLE 3 Graph g(x) = 2 .
x –4
Give the domain and range. Is g(x)
continuous? Is g(x) an odd or even
function?
1
EXAMPLE 3 Graph g(x) = 2 .
x –4
Use reciprocal principles to graph g(x).
1
EXAMPLE 3 Graph g(x) = 2 .
x –4
Use reciprocal principles to graph g(x).
1
EXAMPLE 3 Graph g(x) = 2 .
x –4
D = {x|x ≠ ±2}
R = {y|y  0 or y  -1/4}
g(x) is an even function but is not
continuous.
1
EXAMPLE 4 Graph g(x) = 2 .
x –4
again without graphing its reciprocal
function first.
1. Find the domain excluding values
where the denominator equals zero.
x2 – 4 = 0
x2 = 4
x = ±2
D = {x|x ≠ ±2}
1
EXAMPLE 4 Graph g(x) = 2 .
x –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.
1
EXAMPLE 4 Graph g(x) = 2 .
x –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 xaxis, points within a region will all be
on the same side of the x-axis. Include
the y-intercept as one of the points.
1
EXAMPLE 4 Graph g(x) = 2 .
x –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.
1
EXAMPLE 4 Graph g(x) = 2 .
x –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. Exercises
1. f(x)
►A. Exercises
3. p(x)
►A. Exercises
6. Give the vertical asymptotes of
1
h(x) = 2
x + 5x – 14
►A. Exercises
Evaluate each function as indicated.
1
9. f(x) = 2
for x = 2 and x = -6
x – 25
►B. Exercises
12. Graph the reciprocal function. Give the
domain and range.
1
h(x) = 2
x + 5x – 14
■ Cumulative Review
41. Solve ABC where A = 58°, B = 39°,
and a = 10.5.
■ Cumulative Review
42. Give the period and amplitude of
y = 5 sin 3x.
■ Cumulative Review
43. Find f(4) if
 x2– 8 if x  3
f(x) =  x – 1 if 3  x  9
if x  9
 7x
■ Cumulative Review
44. How many zeros does a cubic
polynomial function have? Why?
■ Cumulative Review
45. 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.