Welcome to a lesson on
graphing the basic rational function,
f of x, equals one divided by x.
We'll also determine the domain and range of the function,
as well as the equations of the asymptotes.
Often the best way to graph a function
we're not familiar with
is to make a table of values.
Typically we begin with input values, or x values,
that are convenient, for example, x equals zero,
one, two and three.
So let's find the outputs, or function values,
for these inputs, or x values.
So when x equals zero,
the function value or function output is f of zero,
so we substitute zero for x,
which would give us one divided by zero,
and division by zero is undefined,
and therefore the function is undefined at x equals zero.
So we'll put Und, for the output,
which means there's no point
in the function at x equals zero, and therefore,
there's a break in the graph at x equals zero.
The next input or x value is one,
so the output or function value is f of one,
which would be one divided by one, which equals one.
This gives us the ordered pair one, one,
notice if the input or x value
is negative one, the function value or output
would be one divided by negative one,
which equals negative one,
giving us the ordered pair negative one, negative one.
Next when the input or x value is two,
the output or function value is f of two,
which is one divided by two, which equals 1/2,
which gives us the ordered pair two, 1/2,
also notice that when the input is negative two,
the output or function value would be
one divided by negative two, or negative 1/2,
giving us the ordered pair negative two, negative 1/2.
When x or the input is three,
the function value or output is f of three,
which is one divided by three which equals 1/3,
giving us the ordered pair three, 1/3.
And when the input or x value is negative three,
the output or function value is
one divided by negative three, or negative 1/3,
giving us the ordered pair of
negative three, negative 1/3.
Let's go ahead and plot these points
on the coordinate plane,
and then we'll come back and complete the table of values.
Let's go ahead and plot these three ordered pairs first,
so the 0.1, one, would be here,
the 0.2, 1/2 is here,
and the 0.3, 1/3 would be here.
Notice how as the inputs or x values increase,
the function values or outputs
are getting smaller and smaller and approaching zero.
But the function value or output will never be zero,
because notice how we have a numerator of one,
which means for the graph,
it's pass through these three points
and approach the function value of zero,
and therefore it will approach the x axis
in this direction here.
Now let's go ahead and plot these three ordered pairs,
so we have negative one, negative one, which is here.
Negative two, negative 1/2, which is here,
negative three, negative 1/3, which is here,
and the same thing,
notice how as the x values
decrease in the negative direction,
the function values or outputs still approach zero,
but this time, from the negative direction,
and therefore we know the graph passes
through these three points and approaches the x axis,
where y equals zero in the negative direction.
Now let's focus on what's happening
to the graph of the function between zero and one,
and between negative one and zero.
Let's select the values of 1/4 and 1/2
for two values between zero and one,
and we'll select the values
of negative 1/2 and negative 1/4,
for two values between zero and negative one.
Let's go ahead and find the output or function value
when the input is 1/2, which would be f of 1/2,
f of 1/2 is equal to one divided by 1/2.
Dividing by one half is the
same as multiplying by the reciprocal,
so this is equal to one times the reciprocal of 1/2,
which is two over one or just two.
One times two is two,
this gives us the ordered pair of 1/2, two,
which means when the input or x value is negative 1/2,
we'd have one divided by negative 1/2
which equals one times negative two,
which equals negative two,
giving us the ordered pair negative 1/2, negative two.
And now let's find the output
or function value when x is 1/4.
The function value is f of 1/4,
which equals one divided by 1/4,
which equals one times 4/1, which equals four,
giving us the ordered pair 1/4, four,
similarly, when x is equal to negative 1/4,
the output or function value would be negative four,
giving us the ordered pair negative 1/4, negative four.
So let's go ahead and plot these
two points on the coordinate plane.
So we have 1/2, two, which is here,
and then 1/4, four, which would be here,
so notice how as the inputs get smaller and smaller,
and approach zero from the positive side,
the function values are getting larger and larger
in the positive direction,
and therefore the graph is going to approach
the positive y axis in this direction here.
So we say the function values are approaching
positive infinity as the inputs or x values
approach zero from the positive side.
And then we have negative 1/2, negative two,
which is here,
and a negative 1/4, negative four, which would be here.
Notice how as the x values approach
zero from the negative side,
the function values are getting smaller and smaller,
in the negative direction approaching negative infinity,
which means the graph approaches
the negative y axis in this direction here.
So this is the graph of the basic function
f of x equals one divided by x.
Notice how the graph approaches the y axis,
and therefore the vertical line, this line here,
that the graph approaches,
is called the vertical asymptote,
and the horizontal line the graph approaches,
which in this case, is the x axis, this line here,
is called the horizontal asymptote.
So to be more formal, an asymptote,
is a line or a curve that the graph
of the function approaches,
or gets closer and closer to.
So we'll give the equations of the
asymptotes in just a minute,
but let's also define the domain and range of a function.
The domain is a set of all possible inputs,
or in this case, all possible x values,
and the range is the set of all possible
outputs or function values, which are often y values.
So let's give the domain/range in equations
of the asymptotes on the next slide.
Here's a computer generated graph
of the function f of x equals one divided by x,
again the vertical asymptote is this
vertical line the graph approaches.
The equation of the axis is x equals zero,
and therefore the vertical asymptote is x equals zero,
the horizontal asymptote is this horizontal line,
the x axis that the graph approaches.
The equation of the x axis is y equals zero.
And then let's give the domain and range,
again, the domain is a set of all
possible inputs or x values.
Remember for the function f of x, equals one divided by x,
x can be any real number except zero,
because when x equals zero, we have division by zero,
which is undefined.
Which means the domain is all reals except zero,
using interval notation,
we can state the domain as the open interval
from negative infinity to zero, union,
the open interval from zero to infinity.
Remember that in parentheses to the right and left
of zero indicates zero is not in the interval.
We could also say x is less than zero,
or x is greater than zero.
Again, these intervals are equivalent,
when it's expressed using interval notation,
and one using inequalities.
And now the range is the set of all
possible outputs or function values,
which are often referred to as y values.
And we also mentioned earlier that
the fraction 1/x will never be equal to zero,
because the numerator is the constant one,
and the only way a fraction can be zero,
is when the numerator is zero
and the denominator is non-zero.
We can also look at graph and see
that the function value is never equal to zero,
which would be a point on the x axis.
So the range is also all real numbers except zero,
so using interval notation,
we would once again have the open interval
from negative infinity to zero, union,
the open interval from zero to infinity.
We could also express the range using inequalities,
using y or f of x, let's go ahead and use f of x,
and say f of x is less than zero,
or f of x is greater than zero.
I hope you found this helpful.