Rational function – Functions – Mathematics – Pre-university Calculus – TU Delft NEW

Rational function – Functions – Mathematics – Pre-university Calculus – TU Delft NEW


Hello. The quotient of two quantities is called their
ratio. A rational function is the quotient,
or ratio, of two polynomials. Let us look at an example. Suppose my trip to work involves 3 km of city
traffic, which always takes me 6 minutes which is 1/10th
of an hour. Subsequently I have to drive 20km on the freeway
with a speed v, which depends on the amount of traffic. Thus the time on the freeway is 20km/v. What will be my average speed? Well,
it is total distance divided by total time. The distance is 23 km. The time spend is 1/10 of an hour for the
city part and 20/v for the part on the freeway. Thus the average speed is 23/(1/10+20/v). Multiplying both numerator and denominator
by v we obtain 23v/(1/10 v+ 20). This rational function is a quotient of two
linear polynomials. In general,
rational functions are precisely those functions you can make with a variable,
numbers, addition,
multiplication and division. These functions can all be written as a fraction
P/Q where P and Q are polynomials. Of course,
Q can not be the polynomial which is always 0. Note that constant functions are also polynomials,
so by taking the denominator to be the constant polynomial Q=1,
we see that any polynomial is also a rational function. Of course those are not the rational functions
that interest us in this video. You do get functions we could not define before,
even in the simple cases where the numerator is constant,
or the denominator is just a power of x. Note that we cannot define a rational function
at a point x where the denominator equals zero. This means that the zeros of the denominator
are not in the domain of the function. You can calculate with rational functions,
just as you are used to with ordinary fractions. In particular if you want to write a sum of
two rational functions as a single fraction, you have to put both terms over a common denominator. For example if we add 1/(x-2) + 1/(x^2+2)
we use (x-2)(x^2+2) as a common denominator. The result is (x^2+x)/[(x-2)(x^2+2)]. Typically factored expressions are easier
to work with, so refrain from expanding the denominator
unless you have a good reason to expand. In general it is not easy to draw the graph
of a rational function. A characteristic feature of graphs of rational
functions are the asymptotes. For example,
the function (2x^2+2x-1)/(x^2+x+1) has a horizontal asymptote for x to infinity. This means that for very large x the graph
approaches a horizontal line very closely, in this case the line y=2. Notice that the graph also approaches this
line for very large negative x, so it is also an asymptote for x to minus
infinity. A rational function has a horizontal asymptote
if the degree of the numerator is at most the degree of the denominator. You can find the location of a horizontal
asymptote for a rational function by dividing both numerator and denominator by the largest
power of x in the denominator. For very large x the terms 1/x,
1/x^2 etcetera tend to zero, so both numerator and denominator become a
constant. In this way you can determine the height of
the horizontal asymptote. A rational function can also have vertical
asymptotes. Consider f(x)=1/(x-2). For x equal to 2 the denominator becomes zero
and the function is not defined. For x slightly more than 2 this becomes 1
over a small positive number, which is very big. For x slightly less than 2 it is 1 over a
small negative number, which is large negative. Therefore the graph has a vertical asymptote
at x=2. A rational function can only have vertical
asymptotes at zeros of the denominator. Now consider g(x)=(x-1)/[(x-1)(x-2)]. This can be simplified to f(x). However g is not defined at x=1,
whereas f is. Apart from this difference in domain the functions
are equal. In the graph,
we denote the fact that 1 is not in the domain by adding an open circle at x=1. Notice that g has no vertical asymptote at
x=1 even though this is a zero of the denominator. This is because the numerator also vanishes
at x=1. Finally consider the graph of x+2+ 1/(x+1). For very large values of x the 1/(x+1) becomes
very small, and the graph of the function thus approaches
the line y=x+2. Similarly for very large negative x. The line y=x+2 is called an oblique asymptote. A rational function can only intersect the
x-axis if the numerator is zero. But beware,
as you saw in a previous example, if numerator and denominator have the same
zero, then this point is not in the domain of our
function. Thus it is not a zero of the rational function. The zeros of a rational function are precisely
those zeros of the numerator for which the denominator is non-zero. The locations of the zeros can heavily influence
what the graph of a rational function looks like. For example compare (x-2)/(x-1)(x-3) to x/(x-1)(x-3). Both graphs have vertical asymptotes at x=1
and x=3, the zeros of the denominator. Since the degree of the numerator is less
than the degree of the denominator both graphs have a horizontal asymptote at y=0. But the first function has a zero in between
the two asymptotes at x=2 and the second function has a zero to the left of either asymptote
at x=0. Therefore the two graphs are quite dissimilar. To summarize:
Rational functions are the quotient of two polynomials. A rational function can only be defined for
the x-values for which the denominator is unequal to zero. You add rational functions by making the denominators
equal. The graph of a rational function often has
asymptotes. Horizontal asymptotes can be found by dividing
both numerator and denominator by the highest power of x occurring in the denominator. Vertical asymptotes can be found by looking
at the zeros of the denominator. The zeros of the rational function are found
at the zeros of the numerator. Good luck with the exercises!

One thought on “Rational function – Functions – Mathematics – Pre-university Calculus – TU Delft NEW

  • June 14, 2018 at 6:05 pm
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    Extremely helpful!

    Reply

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