A rational expression is a fraction that is the quotient of two polynomials. A rational function is defined by a quotient of two polynomial functions.
■ The Reciprocal Function f 1x2 =1x
■ The Function f 1x2=x12
■ Asymptotes
■ Graphing Techniques
■ Rational Models
Rational Function A function ƒ of the form
ƒ1x2 = p1x2 q1x2 ,
where p1x2 and q1x2 are polynomial functions, with q1x2≠0, is a rational function.
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3.5 Rational Functions: Graphs, Applications, and Models
ƒ1x2 = 1
x , ƒ1x2= x + 1
2x2 + 5x- 3 , ƒ1x2 = 3x2 -3x -6
x2 + 8x+ 16 Rational functions Any values of x such that q1x2 = 0 are excluded from the domain of a rational function, so this type of function often has a discontinuous graph—that is, a graph that has one or more breaks in it.
The Reciprocal Function f 1x2 =x1 The simplest rational function with a variable denominator is the reciprocal function.
ƒ1x2 = 1
x Reciprocal function
The domain of this function is the set of all nonzero real numbers. The number 0 cannot be used as a value of x, but it is helpful to find values of ƒ1x2 for some values of x very close to 0. We use the table feature of a graphing calculator to do this. The tables in Figure 41 suggest that ƒ1x2 increases without bound as x gets closer and closer to 0, which is written in symbols as
ƒ1x2 S∞ as xS0.
(The symbol xS0 means that x approaches 0, without necessarily ever being equal to 0.) Because x cannot equal 0, the graph of ƒ1x2= 1x will never intersect the vertical line x= 0. This line is a vertical asymptote.
As x approaches 0 from the left, y1=1x As x approaches 0 from the right, y1=1x approaches -∞. (–1E–6 means approaches ∞.
-1*10-6.)
Figure 41
As x increases without bound (written x S∞), the values of ƒ1x2= 1x get closer and closer to 0, as shown in the tables in Figure 42. Letting x increase without bound causes the graph of ƒ1x2= 1x to move closer and closer to the horizontal line y= 0. This line is a horizontal asymptote.
As x approaches ∞, y1=1x approaches 0 through positive values.
As x approaches -∞, y1=1x approaches 0 through negative values.
Figure 42
The graph of ƒ1x2 = 1x is shown in Figure 43.
Reciprocal Function f 1x2 = 1 x
Domain: 1-∞, 02´ 10, ∞2 Range: 1-∞, 02 ´ 10, ∞2
x y
-2 -12
-1 -1 -12 -2
0 undefined
1
2 2
1 1
2 12
• ƒ1x2 =1x decreases on the open intervals 1-∞, 02 and 10, ∞2.
• It is discontinuous at x = 0.
• The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.
• It is an odd function, and its graph is symmetric with respect to the origin.
–2 1 –2 1
x y
0
f(x) =1 x Vertical asymptote Horizontal
asymptote
−4.1
−6.6
4.1
6.6
f(x) =1 x
Figure 43
−4.1
−6.6
4.1
6.6
y = −2 x
The graph in Figure 44 is shown here using a decimal window. Using a nondecimal window may produce an extraneous vertical line that is not part of the graph.
The graph of y= 1x can be translated and/or reflected.
EXAMPLE 1 Graphing a Rational Function
Graph y = -2x . Give the domain and range and the largest open intervals of the domain over which the function is increasing or decreasing.
SOLUTION The expression -2x can be written as -2 A1xB or 2 A-1xB, indicating that the graph may be obtained by stretching the graph of y =1x vertically by a factor of 2 and reflecting it across either the x-axis or the y-axis. The x- and y-axes remain the horizontal and vertical asymptotes. The domain and range are both still 1-∞, 02´10, ∞2. See Figure 44.
–2 2
–2 2
x y
0
y = – 2x
Figure 44
✔ Now Try Exercise 17.
The graph shows that ƒ1x2 is increasing on both sides of its vertical asymp- tote. Thus, it is increasing on 1-∞, 02 and 10, ∞2.
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3.5 Rational Functions: Graphs, Applications, and Models
EXAMPLE 2 Graphing a Rational Function
Graph ƒ1x2 = x 2+1 . Give the domain and range and the largest open intervals of the domain over which the function is increasing or decreasing.
GRAPHING CALCULATOR SOLUTION
When entering this rational function into the function editor of a calculator, make sure that the numerator is 2 and the denominator is the entire expression
1x +12.
The graph of this function has a vertical asymptote at x = -1 and a horizontal asymptote at y = 0, so it is reasonable to choose a viewing window that contains the locations of both asymptotes as well as enough of the graph to determine its basic characteristics. See
Figure 46. ALGEBRAIC SOLUTION
The expression x 2+1 can be written as 2Qx +11R , indicating that the graph may be obtained by shifting the graph of y = 1x to the left 1 unit and stretching it vertically by a factor of 2. See Figure 45.
The horizontal shift affects the domain, which is now 1-∞, -12´1-1, ∞2. The line x = -1 is the vertical asymptote, and the line y =0 (the x-axis) remains the horizontal asymptote. The range is still 1-∞, 02´10, ∞2. The graph shows that ƒ1x2 is decreasing on both sides of its vertical asymptote.
Thus, it is decreasing on 1-∞, -12 and 1-1, ∞2.
y
1 2
–2 x
x = –1
0
f(x) = 2x + 1
Figure 45
−4.1
−6.6
4.1
6.6
f(x) = 2 x + 1
Figure 46
✔ Now Try Exercise 19.
The Function f 1x2 =x12 The rational function ƒ1x2 = 1
x2 Rational function
also has domain 1-∞, 02 ´ 10, ∞2. We can use the table feature of a graphing calculator to examine values of ƒ1x2 for some x-values close to 0. See Figure 47.
As x approaches 0 from the left, As x approaches 0 from the right, y1=x12 approaches ∞. y1=x12 approaches ∞.
Figure 47
The tables suggest that ƒ1x2 increases without bound as x gets closer and closer to 0. Notice that as x approaches 0 from either side, function values are all posi- tive and there is symmetry with respect to the y-axis. Thus, ƒ1x2S∞ as xS0.
The y-axis 1x= 02 is the vertical asymptote.
As x increases without bound, ƒ1x2 approaches 0, as suggested by the tables in Figure 48. Again, function values are all positive. The x-axis is the horizontal asymptote of the graph.
The graph of ƒ1x2 = x12 is shown in Figure 49.
As x approaches ∞, y1=x12 approaches 0 through positive values.
As x approaches -∞, y1=x12 approaches 0 through positive values.
Figure 48
x y
0 1
1 f(x) = 1
x2
−2.1
−6.6
6.1
6.6
f(x) = 1 x2
Figure 49
Rational Function f 1x2 = 1 x2
Domain: 1-∞, 02´10, ∞2 Range: 10, ∞2
x y
{3 19 {2 14 {1 1 {12 4 {14 16
0 undefined
• ƒ1x2 =x12 increases on the open interval 1-∞, 02 and decreases on the open interval 10, ∞2.
• It is discontinuous at x = 0.
• The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.
• It is an even function, and its graph is symmetric with respect to the y-axis.
EXAMPLE 3 Graphing a Rational Function
Graph g1x2 = 1x +1222 - 1. Give the domain and range and the largest open intervals of the domain over which the function is increasing or decreasing.
SOLUTION The function g1x2= 1x+1222 - 1 is equivalent to g1x2= ƒ1x+ 22 -1, where ƒ1x2 = 1
x2 . This indicates that the graph will be shifted
2 units to the left and 1 unit down. The horizontal shift affects the domain, now 1-∞, -22´1-2, ∞2. The vertical shift af- fects the range, now 1-1, ∞2.
The vertical asymptote has equation x= -2, and the horizontal asymptote has equation y= -1. A traditional graph is shown in Figure 50, with a calculator graph in Figure 51. Both graphs show that this function is increasing on 1-∞, -22 and decreasing on 1-2, ∞2.
✔ Now Try Exercise 27.
–3 –1 1
x x = –2 y
y = –1 4
g(x) = 1 (x + 2)2 0
– 1
This is the graph of y = shifted 2 units to the left and 1 unit down.
1 x2 Figure 50
−2.1
−8.6
6.1
4.6
g(x) = 1 − 1
(x + 2)2
Figure 51
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3.5 Rational Functions: Graphs, Applications, and Models
Asymptotes The preceding examples suggest the following definitions of vertical and horizontal asymptotes.
Asymptotes
Let p1x2 and q1x2 define polynomial functions. Consider the rational func- tion ƒ1x2= pq11xx22 , written in lowest terms, and real numbers a and b.
1. If ƒ1x2 S∞ as xSa, then the line x = a is a vertical asymptote.
2. If ƒ1x2Sb as x S∞, then the line y = b is a horizontal asymptote.
Locating asymptotes is important when graphing rational functions.
• We find vertical asymptotes by determining the values of x that make the denominator equal to 0.
• We find horizontal asymptotes (and, in some cases, oblique asymptotes), by considering what happens to ƒ1x2 as x S∞. These asymptotes determine the end behavior of the graph.
Determining Asymptotes
To find the asymptotes of a rational function defined by a rational expres- sion in lowest terms, use the following procedures.