<< Chapter < Page Chapter >> Page >
In this section, you will:
  • Use arrow notation.
  • Solve applied problems involving rational functions.
  • Find the domains of rational functions.
  • Identify vertical asymptotes.
  • Identify horizontal asymptotes.
  • Graph rational functions.

Suppose we know that the cost of making a product is dependent on the number of items, x , produced. This is given by the equation C ( x ) = 15,000 x 0.1 x 2 + 1000. If we want to know the average cost for producing x items, we would divide the cost function by the number of items, x .

The average cost function, which yields the average cost per item for x items produced, is

f ( x ) = 15,000 x 0.1 x 2 + 1000 x

Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.

In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.

Using arrow notation

We have seen the graphs of the basic reciprocal function and the squared reciprocal function from our study of toolkit functions. Examine these graphs, as shown in [link] , and notice some of their features.

Graphs of f(x)=1/x and f(x)=1/x^2

Several things are apparent if we examine the graph of f ( x ) = 1 x .

  1. On the left branch of the graph, the curve approaches the x -axis ( y = 0 )   as   x .
  2. As the graph approaches x = 0 from the left, the curve drops, but as we approach zero from the right, the curve rises.
  3. Finally, on the right branch of the graph, the curves approaches the x- axis ( y = 0 )   as   x .

To summarize, we use arrow notation    to show that x or f ( x ) is approaching a particular value. See [link] .

Symbol Meaning
x a x approaches a from the left ( x < a but close to a )
x a + x approaches a from the right ( x > a but close to a )
x x approaches infinity ( x increases without bound)
x x approaches negative infinity ( x decreases without bound)
f ( x ) the output approaches infinity (the output increases without bound)
f ( x ) the output approaches negative infinity (the output decreases without bound)
f ( x ) a the output approaches a

Local behavior of f ( x ) = 1 x

Let’s begin by looking at the reciprocal function, f ( x ) = 1 x . We cannot divide by zero, which means the function is undefined at x = 0 ; so zero is not in the domain . As the input values approach zero from the left side (becoming very small, negative values), the function values decrease without bound (in other words, they approach negative infinity). We can see this behavior in [link] .

x –0.1 –0.01 –0.001 –0.0001
f ( x ) = 1 x –10 –100 –1000 –10,000

We write in arrow notation

as  x 0 , f ( x )

As the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). We can see this behavior in [link] .

x 0.1 0.01 0.001 0.0001
f ( x ) = 1 x 10 100 1000 10,000

We write in arrow notation

As  x 0 + ,   f ( x ) .

See [link] .

Graph of f(x)=1/x which denotes the end behavior. As x goes to negative infinity, f(x) goes to 0, and as x goes to 0^-, f(x) goes to negative infinity. As x goes to positive infinity, f(x) goes to 0, and as x goes to 0^+, f(x) goes to positive infinity.

This behavior creates a vertical asymptote , which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line x = 0 as the input becomes close to zero. See [link] .

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 5

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask