Pre-Calculus 11 Chapter 7 Absolute Value and Reciprocal Functions.
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Lesson Notes 7.4: Reciprocal Functions.
Objectives:
• graphing the reciprocal of a given function
• analysing the graph of the reciprocal of a given function
• comparing the graph of a function to the graph of the reciprocal of that function
1
• identifying the values of x for which the graph of y =
has vertical asymptotes
f ( x)
Perhaps you have travelled to Mexico or to Hawaii and have exchanged
Canadian dollars for pesos or U.S. dollars. Perhaps you have travelled overseas and
exchanged British pounds for the Japanese yen or Swiss franc. If so, you have
experienced exchange rates in action. Do you know how they work?
Suppose you are taking a trip to Hong Kong and if the Canadian dollar is worth
HK$7.00, it costs C$0.1429 to buy HK$1.00. Change the values 7.00 and 0.1429 to
fractions in lowest terms. Can you see how these fractions are related to each other?
1) How many HK$ can you purchase with CAN$80? With CAN$220?
2) How many CAN$ can you purchase with HK$80? With HK$220?
Recall that the product of a number and its reciprocal is always equal to 1.
3
4
4
3
3 4
is the reciprocal of and is the reciprocal of because ( ) = 1.
4
3
3
4
4 3
1
1
So, for any non-zero real number a, the reciprocal of a is and the reciprocal of is a.
a
a
1
For a function f (x), its reciprocal function is
, provided that f (x) ≠ 0.
f ( x)
For example,
The reciprocal function is not defined when the denominator is 0, so f (x) ≠ 0.
Let us take a look at some simple examples.
Example 1) If y = x + 1, or f (x) = x + 1.
a) graph f (x).
b) find the x-intercept and y-intercept.
c) state the domain and range.
Now, let us set up a reciprocal function of f (x), which is represented as y =
where f (x) = x + 1, y =
a) graph
1
. Examine how the functions are related.
x 1
1
.
f ( x)
b) find the x-intercept and y-intercept.
c) state the domain and range.
1
,
f ( x)
The function y = x + 1 is a linear function of degree one, so its graph is a straight
1
line. The function y =
is a rational function. Its graph has two distinct pieces, or
x 1
branches. These branches are located on either side of the vertical asymptote, defined by
the non-permissible value of the domain of the rational function, and the horizontal
asymptote, defined by the fact that the value 0 is not in the range of the function.
asymptote
• a line whose distance from a given curve approaches zero
vertical asymptote
• for reciprocal functions, occur at the non-permissible values of the function
• the line x = a is a vertical asymptote if the curve approaches the line more and more
closely as x approaches a, and the values of the function increase or decrease without
bound as x approaches a
horizontal asymptote
• describes the behavior of a graph when |x| is very large
• the line y = b is a horizontal asymptote if the values of the function approach b when |x|
is very large.
Example 2) Consider f (x) = 2x + 5.
a) graph f (x).
b) find the x-intercept and y-intercept.
c) state the domain and range.
d) determine its reciprocal function y =
1
.
f ( x)
e) determine the equation of the vertical asymptote of the reciprocal function.
f) graph the reciprocal function y =
1
.
f ( x)