Question Video: Determining the Domain and Range of a Quadratic Function from Its Graph | Nagwa Question Video: Determining the Domain and Range of a Quadratic Function from Its Graph | Nagwa

# Question Video: Determining the Domain and Range of a Quadratic Function from Its Graph Mathematics • Second Year of Secondary School

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Determine the domain of the function represented in the graph. What is the range of the function?

03:54

### Video Transcript

Determine the domain of the function represented in the graph. What is the range of the function?

In this question, we’re given the graph of a function, and we need to determine its domain and range. Let’s start by recalling what we mean by the domain of a function. The domain of any function is the set of all input values for that function. And we need to determine this set by using the graph of the function. And to do this, we need to recall when we sketch the graph of a function, the 𝑥-coordinate of any point on our curve tells us the input value for our function and the 𝑦-coordinate tells us the output value. For example, we can see that our curve passes through the point with coordinates three, two. The 𝑥-coordinate tells us the input to the function, and the 𝑦-coordinate tells us the output. So 𝑓 evaluated at three is equal to two. In particular, this tells us that three is in the domain of our function.

Another way of thinking about this is to consider the vertical line 𝑥 is equal to three. We can see there’s one point of intersection between this vertical line and our function. Because there’s one point of intersection between the vertical line and our curve, there’s one point on the curve of our function with 𝑥-coordinate three. So it must be in the domain of our function. We can do this with other values of 𝑥. For example, if we sketch the vertical line 𝑥 is equal to negative one, we can see there’s a single point of intersection between our curve and the vertical line. Therefore, negative one is also in the domain of our function.

We can notice any vertical line between negative one and three will intersect our curve once. And in fact, since our curve is going to continue infinitely in either direction, any vertical line will intersect the curve once. And therefore, all values of 𝑥 will be in the domain of our function. The domain of the function given in the graph is the set of real numbers ℝ.

Let’s clear some space and move on to the second part of our question. We need to determine the range of this function. And we can recall the range of any function is the set of all output values for the function given the domain or the set of input values. And we can find the range of this function from its graph in a very similar way.

Remember, the 𝑥-coordinate of any point on our curve tells us a possible input value of our curve. And the corresponding 𝑦-coordinate tells us the output for that value of 𝑥. For example, because our curve passes through the point with coordinates two, three, we can conclude our function evaluated at two must be equal to three. Three is in the range of our function.

Therefore, another way of thinking of the range of a function is the set of all 𝑦-coordinates of points which lie on its curve. And we can determine this from its graph. For example, we can see the line 𝑦 is equal to three intersects our curve once. And because there’s a point of intersection, there is a point on the curve with 𝑦-coordinate three. So three is in the range of our function. And we can try this with other values of 𝑦 as well. For example, we can sketch the horizontal line 𝑦 is equal to negative two. This time, we can see there’s two points of intersection between our line and our curve. And it doesn’t matter that there’s more than one point of intersection because all we’re interested in is negative two, a possible output of our function. And the answer is yes, we found two possible input values of 𝑥 which give us an output of negative two. So negative two is in the range of our function.

However, this is not true for all values of 𝑦. For example, if we sketch the line 𝑦 is equal to five, we can see there’s no point of intersection between our curve and line. Graphically, we can see why this is true. This is because three is the largest output of our function. And because the graph of our function continues towards negative ∞ in both directions, our function is unbounded. So any value less than or equal to three will be in the range of our function. And we can write this range in set notation, where we need to use a square parenthesis on the right side of this expression because three is in the range of our function.

This then gives us our final answer. The domain of the function represented in the graph is the set of all real numbers. And the range of the function is the set of all real numbers less than or equal to three.

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