Video Transcript
Consider the function 𝑓 of 𝑥 is equal to the square root of negative 𝑥. Find the domain of 𝑓 of 𝑥. Find the range of 𝑓 of 𝑥.
We begin by reminding ourselves what we actually mean when we talk about the domain and range of a function. The domain of a function is the set of all possible inputs to that function that will make the function work and will output real 𝑦-values. Then, once we have that set and we substitute those values into the function, the range is the set of possible outputs that we get. So thinking about our function 𝑓 of 𝑥 is equal to the square root of negative 𝑥, we’ll use the fact that the domain and range of a function the square root of 𝑥 is the left-closed, right-open interval from zero to ∞. In other words, the 𝑥-values that we substitute into this function must be nonnegative.
More generally, if we have a composite square root function, the square root of 𝑔 of 𝑥, the domain can be found by calculating the values of 𝑥 that satisfy 𝑔 of 𝑥 is greater than or equal to zero. In the case of our function then, we say that this is values such that negative 𝑥 is greater than or equal to zero. If we multiply through by a negative number, in this case, negative one, we need to reverse the inequality symbol. So multiplying through by negative one and we find 𝑥 must be less than or equal to zero. Using set notation then, we can say that the domain of 𝑓 of 𝑥 is the left-open, right-closed interval from negative ∞ to zero.
Now that we have the domain of 𝑓 of 𝑥, we’re able to find the range. We said that the range of the square root of 𝑥 is the left-closed, right-open interval from zero to ∞. In other words, for any number 𝑦 in this interval, we can find some number 𝑥 that satisfies the equation 𝑦 is equal to the square root of 𝑥. This means that if we choose a number negative 𝑥, it will satisfy 𝑓 of negative 𝑥 is equal to the square root of negative negative 𝑥. That’s the square root of positive 𝑥, which is equal to 𝑦. So any number in the left-closed, right-open interval from zero to ∞ is a possible function value of 𝑓 of 𝑥. So the range is the left-closed, right-open interval from zero to ∞.
Now we answered this algebraically, but we could have answered it graphically. Consider the graph of 𝑦 is equal to the square root of 𝑥. Since the domain is the set of values we can substitute into the function, we can think about it on the graph as the spread of values in the horizontal direction. And we see that this is values of 𝑥 greater than or equal to zero. If we then consider the graph of 𝑦 equals the square root of negative 𝑥, we know that we can map 𝑦 is equal to the square root of 𝑥 onto this graph by reflecting it in the 𝑦-axis.
And so we see that the domain flips. It’s all values of 𝑥 less than or equal to zero. And we can also now see why the range remains unchanged. Since the range is the set of possible outputs, we think about it as the spread of values in the 𝑦-direction. In both cases, this is 𝑦-values greater than or equal to zero. So the range once again is the left-closed, right-open interval from zero to ∞.
