WEBVTT
00:00:01.720 --> 00:00:08.920
Which of the following is the range of the function π of π₯ is equal to the absolute value of π₯ plus eight minus seven?
00:00:10.080 --> 00:00:16.480
Is it option (A) the set of real numbers minus the set containing negative seven?
00:00:17.400 --> 00:00:19.960
Option (B) the set of real numbers.
00:00:20.880 --> 00:00:24.920
Option (C) the open interval from negative seven to β.
00:00:25.880 --> 00:00:39.360
Is it option (D) the left-closed, right-open interval from negative seven to β or option (E) the set of real numbers minus the set containing negative eight?
00:00:41.360 --> 00:00:45.800
In this question, weβre asked to determine the range of a given function.
00:00:46.440 --> 00:00:54.960
And we can see the given function π of π₯ contains the absolute value function, and weβre given a graph of this function.
00:00:55.720 --> 00:01:01.040
This means we can answer this question algebraically by looking at the equation of the function.
00:01:01.800 --> 00:01:09.680
Or we can answer the question graphically by considering the meaning of the range with respect to the graph of the function.
00:01:10.960 --> 00:01:13.600
Letβs use the latter method.
00:01:14.360 --> 00:01:22.640
Itβs the set of all output values of the function given its domain, which is the set of possible input values.
00:01:24.000 --> 00:01:27.640
Letβs attempt to determine the range of this function from its graph.
00:01:28.640 --> 00:01:44.080
To do this, we know the π₯-coordinate of a point on the graph tells us the input value of the function and the corresponding π¦-coordinate tells us the output of the function.
00:01:45.240 --> 00:01:54.600
For example, we can see the graph of this function passes through the point with coordinates negative 30, 15.
00:01:55.560 --> 00:02:00.640
Therefore, π evaluated at negative 30 must be equal to 15.
00:02:01.680 --> 00:02:05.520
15 is an element of the range of this function.
00:02:06.080 --> 00:02:08.480
Itβs a possible output.
00:02:09.000 --> 00:02:14.760
And we can, of course, verify this by substituting negative 30 into the function π of π₯.
00:02:16.120 --> 00:02:20.720
But the range of our function is the set of all possible output values of the function.
00:02:21.400 --> 00:02:36.600
Since the π¦-coordinates of points on the graph tells us the possible outputs of the function, the range of the function is the set of all π¦-coordinates of points on its graph.
00:02:37.760 --> 00:02:43.720
So letβs try and determine the π¦-coordinates of points which lie on the graph.
00:02:45.360 --> 00:02:51.120
To do this, we can note that there is a point with lowest π¦-coordinate.
00:02:52.000 --> 00:02:54.840
Itβs the point which lies right on the corner.
00:02:55.680 --> 00:03:05.840
However, we cannot determine the π¦-coordinate of this point just from the diagram, so weβll need to determine the coordinates of this point by using the function.
00:03:07.360 --> 00:03:12.280
To do this, we recall π of π₯ is the absolute value of π₯ plus eight, and then we subtract seven.
00:03:13.200 --> 00:03:19.600
And the point with lowest π¦-coordinate will be when the output value of this function is the lowest.
00:03:20.560 --> 00:03:26.720
So we need to make the absolute value of π₯ plus eight minus seven as small as possible.
00:03:27.880 --> 00:03:36.520
To do this, we note that the absolute value of π₯ plus eight will always be greater than or equal to zero.
00:03:37.640 --> 00:03:42.160
And we canβt affect the value of negative seven since itβs a constant.
00:03:42.920 --> 00:03:50.360
Therefore, the smallest output of this function will be when the absolute value of π₯ plus eight is equal to zero.
00:03:51.320 --> 00:03:54.200
This occurs when π₯ is negative eight.
00:03:54.960 --> 00:04:02.960
Therefore, weβve shown that negative eight is the input value of the lowest output of our function.
00:04:03.920 --> 00:04:09.560
Negative eight is the π₯-coordinate of the corner.
00:04:10.560 --> 00:04:13.680
And we can use this to determine the π¦-coordinate.
00:04:14.400 --> 00:04:17.920
The π¦-coordinate will be π evaluated at negative eight.
00:04:18.760 --> 00:04:24.400
And we can evaluate our function at negative eight by substituting negative eight into the function.
00:04:25.080 --> 00:04:30.040
Itβs the absolute value of negative eight plus eight minus seven.
00:04:30.760 --> 00:04:32.240
And this is equal to negative seven.
00:04:33.000 --> 00:04:41.600
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.
00:04:42.120 --> 00:04:46.240
Itβs the smallest element of the range.
00:04:47.400 --> 00:04:56.920
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.
00:04:57.800 --> 00:05:08.240
And in particular, this means for any π¦-value greater than or equal to negative seven, there is a point on the graph of the function with this as a π¦-coordinate.
00:05:09.080 --> 00:05:13.400
In other words, itβs a possible output of the function.
00:05:14.280 --> 00:05:20.280
Therefore, the range of this function includes negative seven and is unbounded.
00:05:21.320 --> 00:05:22.880
It goes all the way to β.
00:05:23.560 --> 00:05:31.400
We write this as the left-closed, right-open interval from negative seven to β.
00:05:32.320 --> 00:05:47.400
Hence, we were able to show the range of the function π of π₯ is equal to the absolute value of π₯ plus eight minus seven is option (D): the left-closed, right-open interval from negative seven to β.