We want to determine the domain of the following rational functions.
To determine the domain of a rational function, we want to find the x values that we must exclude
from the domain. Since a fraction bar represents division and we know that division by zero is
undefined, to find the domain of a rational function, we want to set the denominator equal
to zero and solve for x. This will tell us which values we must exclude from the domain.
So for our first function, we want to set two x minus five equal to zero and solve for x.
So, we’d add five to both sides of the equation. That would give us two
x equals five. Then divide both sides by two. So this tells us that x equals five halves
would make the denominator equal to zero, so we must exclude five halves from the domain.
So the domain would be all real numbers, or we can say all reals except
x equals five halves. If we were to graph the domain, we would graph the number line,
plot five halves, and exclude this value. So we’d have an open point here,
and then we’d shade every other real number. So we’d shade to the right and to the left.
If we wanted to express this using interval notation, we’d be approaching positive infinity
on the right and negative infinity on the left. So, using interval notation, we’d
have the interval from negative infinity to five halves, union five halves to positive infinity.
So here we express the domain in words,
here we express it as a graph, and here we express it using interval notation.
For our second function, we need to set two x squared plus three x equal to zero
and solve for x to determine the excluded values. So we’d have the equation two x squared
plus three x equals zero. The first step in factoring is to factor out the greatest
common factor, which is x. So I have x times the quantity two x plus three equals zero.
So either x equals zero from this first factor or two x plus three equals zero.
Well, here we would subtract three and divide by two, so we’d have x equals negative three halves.
So we must exclude zero and negative three halves from the domain.
So the domain would be all real numbers; we can abbreviate real numbers by using this symbol here;
it’s like a two-legged r, except x equals negative three halves and zero. So again, if we wanted to
graph the domain, we would sketch a number line and exclude zero and negative three halves.
So let’s say zero is here and negative three halves is here. We’d exclude these two values
and graph every other real number. So we’d graph in between to the right and to the left.
If we wanted to express this using interval notation, this would be negative infinity;
this would be positive infinity. So we’d have from negative infinity to negative three halves,
union from negative three halves to zero, union from zero to infinity.
For the last example, we take the denominator of x squared plus four x minus twenty-one equal to zero
and solve for x to determine the excluded values.
So we’d have x squared plus four x minus twenty-one equals zero.
This does factor, so the first terms would be the factors of x squared, which would be x and x.
Now we want the factors of negative twenty-one that add to positive four.
That would be positive seven and negative three. So the solutions to this equation are x equals
negative seven or x equals positive three, which means these values make
the denominator equal to zero and must be excluded from the domain.
So the domain would be all real numbers, or all reals except
x equals negative seven and three. We’ll graph the domain as well.
So we would plot negative seven and positive three, exclude these values with open points,
and graph every other real number. So we graph in the middle to the right and to the left.
Using interval notation, on the left, we’d have the interval from negative
infinity to negative seven, union negative seven to three, union three to infinity.
Now there is one more thing I’d like to mention. Sometimes rational functions will simplify
because they have common factors between the numerators and denominators,
but we do not want to simplify them before determining the domain and range. What I mean
by that is if we take a look at g of x, we could write this as g of x equals the numerator of x,
and the denominator in factored form would be x times the quantity two x plus three.
So we do not want to simplify out this common factor of x because then we would not
exclude zero from the domain. So this common factor does produce a hole in the function,
while the zero of this factor produces a vertical asymptote. So it does affect the graph,
but we need all of the factors of the denominator to determine the domain.
Okay, I hope you found this helpful.