Video Transcript
Determine the domain and range of
the function 𝑓 of 𝑥 is equal to one divided by 𝑥 minus five.
The question wants us to determine
the domain and range of the function 𝑓 of 𝑥. And we can see that our function 𝑓
of 𝑥 is a rational function. It’s the quotient of two
polynomials. We can also see we’re given a graph
of the function 𝑓 of 𝑥. Let’s start by finding the domain
of this function by using the graph. To start, remember, the domain is
all the inputs of our function.
To find all the possible inputs of
our function 𝑓 of 𝑥, let’s take a look at the values of 𝑥 our function can
take. We want to find the values of 𝑥
where our function is undefined. We can see, for example, when 𝑥 is
equal to six, our function outputs the value of one. So we can see six is in the domain
of our function 𝑓 of 𝑥. In fact, there’s only one value
where our function 𝑓 of 𝑥 is undefined.
If we draw the vertical line 𝑥 is
equal to five, we can see that our function 𝑓 of 𝑥 does not intersect this
line. This means our function is not
defined when 𝑥 is equal to five. Every other vertical line will
intersect our function, so this is the only point where our function is not
defined. So we’ve shown our function 𝑓 of
𝑥 is defined everywhere except where 𝑥 is equal to five. In other words, the domain of 𝑓 of
𝑥 is the real numbers minus the point where 𝑥 is equal to five.
We now need to find the range of
our function. Remember, the range of our function
is the set of all possible outputs of our function. We can do something very similar to
check whether a value is in the range of our function. For example, to check whether
negative one is in the range of our function, we draw a horizontal line from
negative one to the curve and then see the value of 𝑥 which gives us this
output. We see from the graph when 𝑥 is
equal to four, our function outputs negative one. Therefore, negative one is in the
range of our function. We can see all horizontal lines
will intercept our function except the one when 𝑦 is equal to zero. The line 𝑦 is equal to zero does
not intercept our curve. In other words, no value of 𝑥
outputs the value of zero.
So for our function 𝑓 of 𝑥,
there’s a value of 𝑥 which outputs every number except the value of zero. In other words, the range of 𝑓 of
𝑥 is the real numbers minus the point zero. Therefore, given a graph of the
function 𝑓 of 𝑥 is equal to one divided by 𝑥 minus five, we were able to show the
domain of this function is all real numbers except when 𝑥 is equal to five and the
range of this function is all real numbers except for zero.
One important caveat of this
example is that we can find the domain of our rational function by finding the
vertical asymptotes and we can find the range of our function by finding the
horizontal asymptotes.
