Question Video: Determining the Domain and Range of a Rational Function given Its Graph Mathematics

Determine the domain and range of the function 𝑓(π‘₯) = 1/(π‘₯ βˆ’ 5).

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Video Transcript

Determine the domain and range of the function 𝑓 of π‘₯ is equal to one divided by π‘₯ minus five.

The question wants us to determine the domain and range of the function 𝑓 of π‘₯. And we can see that our function 𝑓 of π‘₯ 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 π‘₯. 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 π‘₯, let’s take a look at the values of π‘₯ our function can take. We want to find the values of π‘₯ where our function is undefined. We can see, for example, when π‘₯ 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 π‘₯. In fact, there’s only one value where our function 𝑓 of π‘₯ is undefined.

If we draw the vertical line π‘₯ is equal to five, we can see that our function 𝑓 of π‘₯ does not intersect this line. This means our function is not defined when π‘₯ 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 π‘₯ is defined everywhere except where π‘₯ is equal to five. In other words, the domain of 𝑓 of π‘₯ is the real numbers minus the point where π‘₯ 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 π‘₯ which gives us this output. We see from the graph when π‘₯ 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 π‘₯ outputs the value of zero.

So for our function 𝑓 of π‘₯, there’s a value of π‘₯ which outputs every number except the value of zero. In other words, the range of 𝑓 of π‘₯ is the real numbers minus the point zero. Therefore, given a graph of the function 𝑓 of π‘₯ is equal to one divided by π‘₯ minus five, we were able to show the domain of this function is all real numbers except when π‘₯ 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.

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