Question Video: Finding the Range of a Function from Its Graph Mathematics • Second Year of Secondary School

Video Transcript

Which of the following is the range of the function 𝑓 of 𝑥 is equal to the absolute value of 𝑥 plus eight minus seven? Is it option (A) the set of real numbers minus the set containing negative seven? Option (B) the set of real numbers. Option (C) the open interval from negative seven to ∞. Is it option (D) the left-closed, right-open interval from negative seven to ∞ or option (E) the set of real numbers minus the set containing negative eight?

In this question, we’re asked to determine the range of a given function. And we can see the given function 𝑓 of 𝑥 contains the absolute value function, and we’re given a graph of this function. This means we can answer this question algebraically by looking at the equation of the function. Or we can answer the question graphically by considering the meaning of the range with respect to the graph of the function. Let’s use the latter method. It’s the set of all output values of the function given its domain, which is the set of possible input values.

Let’s attempt to determine the range of this function from its graph. To do this, we know the 𝑥-coordinate of a point on the graph tells us the input value of the function and the corresponding 𝑦-coordinate tells us the output of the function. For example, we can see the graph of this function passes through the point with coordinates negative 30, 15. Therefore, 𝑓 evaluated at negative 30 must be equal to 15. 15 is an element of the range of this function. It’s a possible output. And we can, of course, verify this by substituting negative 30 into the function 𝑓 of 𝑥.

But the range of our function is the set of all possible output values of the function. Since the 𝑦-coordinates of points on the graph tells us the possible outputs of the function, the range of the function is the set of all 𝑦-coordinates of points on its graph. So let’s try and determine the 𝑦-coordinates of points which lie on the graph. To do this, we can note that there is a point with lowest 𝑦-coordinate. It’s the point which lies right on the corner. However, we cannot determine the 𝑦-coordinate of this point just from the diagram, so we’ll need to determine the coordinates of this point by using the function.

To do this, we recall 𝑓 of 𝑥 is the absolute value of 𝑥 plus eight, and then we subtract seven. And the point with lowest 𝑦-coordinate will be when the output value of this function is the lowest. So we need to make the absolute value of 𝑥 plus eight minus seven as small as possible. To do this, we note that the absolute value of 𝑥 plus eight will always be greater than or equal to zero. And we can’t affect the value of negative seven since it’s a constant. Therefore, the smallest output of this function will be when the absolute value of 𝑥 plus eight is equal to zero. This occurs when 𝑥 is negative eight. Therefore, we’ve shown that negative eight is the input value of the lowest output of our function. Negative eight is the 𝑥-coordinate of the corner.

And we can use this to determine the 𝑦-coordinate. The 𝑦-coordinate will be 𝑓 evaluated at negative eight. And we can evaluate our function at negative eight by substituting negative eight into the function. It’s the absolute value of negative eight plus eight minus seven. And this is equal to negative seven. Therefore, negative seven is an element of the range of our function and no value smaller than negative seven is an element of the range. It’s the smallest element of the range.

To determine the rest of the range of this function, we need to recall that the graph of this function continues indefinitely in both directions. And in particular, this means for any 𝑦-value greater than or equal to negative seven, there is a point on the graph of the function with this as a 𝑦-coordinate. In other words, it’s a possible output of the function. Therefore, the range of this function includes negative seven and is unbounded. It goes all the way to ∞. We write this as the left-closed, right-open interval from negative seven to ∞. Hence, we were able to show the range of the function 𝑓 of 𝑥 is equal to the absolute value of 𝑥 plus eight minus seven is option (D): the left-closed, right-open interval from negative seven to ∞.