Find the domain and the range of the function 𝑓 𝑥 is equal to 𝑥 plus one all squared minus two.
We’re gonna be able to find the domain and the range because we’ve been given the graph here. We’re gonna do that using that graph. First, we will need to know what is the domain. So one way of thinking about the domain is that it is all possible input values. So that means all 𝑥-values in parentheses. So let’s have a look at the graph.
Now, as you can see, I’ve chosen some arbitrary 𝑥-values, just going to highlight what domain could be. So I’ve picked negative three, negative two, one, and two — all values of 𝑥. If I draw a line either straight up or straight down, you can see that it will intersect with our curve. And in fact, as we’ve got these two arrows on our graph, this tells us that actually our graph extends infinitely. So our graph will keep going. So and because of that, we can actually see that actually any 𝑥-value will work. So we can say that domain of a function is equal to all of our real values. And we represented it using this notation here. This is gonna be capital 𝑅.
Great! So we now know the domain of our function. We’re now gonna try to find the range. Our range is a set of possible output values. So that really means is our 𝑦-values that are possible. Okay, so let’s have a look at the graph and see if this can help us with this.
First of all, I’ve chosen a couple of arbitrary values here. So I’ve chosen 𝑦 is equal to seven, 𝑦 is equal to four. And you can see that both of those are lines would intersect with our graph. And they’ll do that on both sides actually. So also because we’ve got the arrow, which we highlighted earlier, we see that this graph extends up to infinity. So it keeps going upwards. It means that our 𝑦-value can actually be any value up to infinity.
However, if we then move down the graph, we’ll see that it is a different story when we get to the lowest point of the graph. We can see that actually at negative two, there is a possible output value because we can see that actually the graph’s minimum point is there. So we’ll have our 𝑥-value of negative one when our 𝑦-value was negative two. However, where I’ve drawn the other line at negative six, we see that actually there wouldn’t be any possible output values. So therefore, we can say that our 𝑦-values could be anything down to and including negative two.
And now that we found the range, we can actually say that we’ve solved the problem because the domain of this function is all real roots. And the range of the function is anything from and including negative two all the way up to infinity. And that’s the 𝑦-values.
And we have it in this notation cause this is called interval notation. And we use the bracket on the left-hand side because it means that it is down to and including negative two. And that’s part of our interval notation. And on the right-hand side, we use the parentheses. And that is because our value cannot include infinity, but it means that it goes 𝑦-values up to infinity.