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18

RATIONAL FUNCTIONS

Singularities

A RATIONAL FUNCTION is a quotient of polynomials.  It will look like this:

y  =   g(x)
h(x)
,

where g and h are polynomials (h not equal0).

A rational function will have an x-intercept -- y will equal 0 -- only if the numerator g(x) = 0.  (Lesson 5 of Algebra.)

Consider

y  =       1   
x − 8
.

Since the numerator 1 will never be 0, the graph of that function never touches the x-axis.

 Now a denominator may not be 0.  The symbol  1
0
 has no meaning.
  Therefore, in the rational function       1   
x − 8
 ,  x may not have the value 8.

8 is called a singularity of that function.

A singularity of a function is any value of the variable that would make a denominator 0.

Problem 1.   In each of the following, which values of x may not be included in the domain?  That is, which values are the singularities of the function?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

  a)  y 1
x
   x = 0     b)  y     1   
x + 5
   x = −5     c)  y     1    
2x + 1
   x = −½
  d)  y      1     
x2 − 16
   x = 4 and −4     e)  y         1       
x2 + x − 6
   x = −3 and 2

The reciprocal function

In Topic 8 we saw the graph of the reciprocal function,

y = f(x) =   1
x

Rational function

That is also the equation of a hyperbola, which, like the parabola, is one of the conic sections.

But say that we did not know how to draw the graph.  Then we would ask -- and answer -- the following questions.  The student should understand that these same questions must be answered when drawing the graph of any rational function.

1.  Is the graph symmetrical with respect to the y-axis or to the
1.   origin?

Answer.   We must look at  f(−x):
f(−x)  =     1  
x
  =  −  1
x
  =  − f(x).

Since f(−x) = −f(x), the graph is symmetrical with respect to the origin.  Hence, we need only draw the right-hand side.  The left side will follow from the symmetry.

2.  Are there any x- or y-intercepts?

Can we solve  y  =   1
x
  =  0 ?

No, we cannot, because the numerator is not 0. (Lesson 5 of Algebra.) Therefore there are no x-intercepts.

And x may not be 0. Therefore, there are no y-intercepts, either.

3.  What happens to the value of y when x is very large?

y =   1
x

If x is a very large positivenumber, then y will be a very small positive number.  This means that at the extreme right end of the x-axis,

Rational function

the height of the graph -- the value of y -- will be very close to the x-axis, and on the positive side:

4.  Where is the singularity of this function?

At x = 0.

5.  What happens to the value of y when x is close to a
5.   singularity?

y becomes extremely large:

Rational function

When x is close to 0 in this case -- when x is a very small positive number -- then y, its reciprocal, will be a very large positive number.

Now, we know that there are no intercepts.  And x will take every value greater than 0.  Therefore, those two ends of the graph must

  connect.  Here is the graph of  y =  1
x
 :

Rational function

Asymptotes

Briefly, an asymptote is a straight line that a graph comes closer and closer to but never touches.

The x-axis is a horizontal asymptote of that graph.  The graph and the x-axis come closer and closer but never touch.

More precisely:

Rational function

"The horizontal line  y = b  is a horizontal asymptote of the graph of
y = f(x)"  means:  Upon naming any positive number, however small, it will be possible to name a value of f(x -- call it f(x0) -- such that the distance between f(x0) and b -- and all subsequent values of |f(x) − b| -- will be smaller than that small number.

In other words, the distance between the graph and the horizontal asymptote becomes almost 0.

The y-axis is a vertical asymptote of that graph.

It is true that the distance between the graph and the vertical axis becomes almost 0.  But what it is important is to see that, as a graph approaches a vertical asymptote, its absolute value becomes extremely large.

Rational function

"The vertical line  x = a  is a vertical asymptote of the graph of
y = f(x)"  means:  Upon naming any positive number, however large, it will be possible to name a value of x close to a -- call it x0 -- such that the absolute value of f(x0), and all values with x closer to a, will be larger than that large number.

Thus at values of x close to a singularity -- when the denominator is an extremely small number, almost 0 -- the values of the function become extremely large.

Problem 2.   Where do we always find a vertical asymptote of a graph?

At a singularity.

Problem 3.   What does the equation of a vertical line look like?

x = A number.

See Lesson 33 of Algebra.

Problem 4.   Write the equation of the vertical asymptote(s) of each of the following.

  a)  y 1
x
   x = 0     b)  y     1   
x + 5
   x = −5     c)  y     1    
2x + 1
   x = −½
  d)  y      1     
x2 − 16
   x = 4  and  x = −4
 
  e)  y         1       
x2 + x − 6
   x = −3  and   x = 2

Problem 5.   Each of the following is a translation or transformation of

    y 1
x
.  Sketch the graph of each one.
  a)   y    1    
x − 3

Rational function

  b)   y    1    
x + 2

       Rational function

  c)   y = −     1    
x + 2
       Rational function

This is a reflection about the x-axis of graph b).

  d)   y    1    
x − 3
  + 2

Rational function

End of the lessson

Next Topic:  Inverse functions


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