Use the vertical line test to determine whether or not a graph represents a function. If a vertical line is moved across the graph and, at any time, touches the graph at only one point, then the graph is a function. If the vertical line touches the graph at more than one point, then the graph is not a function. Show
One of the neat things about functions is that we know something is a function if each x has exactly one y but sometimes you are not given the points, you are not given numbers, all you are given is a funny graph. So what I want to talk about here is how you can tell that something is a function just based on the graph and you'll see it's actually pretty easy. It's using what's called the vertical line test. So what I'm going to do is go through these graphs and draw vertical lines and if it hits, if my vertical line hits the graph more than once in each line then it's not a function, because that represents a place where an x value has two y values. Let's check it out. Think of this like your pencil this is a big pencil. What you would do with the graph on your paper is take your pencil lay it down there and then move it across the graph, see if you hit any places on this graph where your pencil crosses the squiggly in more than one place. And you'll see in this graph there's like tons of places, check it out. I just hit my graph like one two three four like 10 times whatever it doesn't matter, I hit it more than once so this is not a function. This x value right here whatever it is has tons the y values there is a y value, there is another one, there is no one it's not a function. Each x only gets one y value. Let's try this next graph use your pencil and make sure it's vertical and not horizontal. Vertical oh-oh! You can see your pencil hits places where your vertical line crosses the graph in more than one place. That again means that x has two y values, not a function. Here is a couple that are a little bit different when you use your vertical line test down here. Check it out everywhere I move my pen it only crosses the graph once right, I'm never hitting this graph line more than once. So this case yes it's a function because that x value only has one y value. Very similar here when I use my pen and move it vertically across the graph there is nowhere where I'm hitting the shape twice, I'm only hitting it once therefore d, yes, is also a function. If you don't remember anything else from this video what I hope you remember is the vertical line test. If a graph passes the vertical line test then it is a function. What I mean by that is if you move your pen and it hits only once then yes it's a function, if it hits more than once, no it's not a function. I personally kind of like theses problems I think they are not too hard and there is no numbers involved so that's kind of cool. Image transcription text 2.5.3 Test (CST): Functions Does this graph show a function? Explain how you know. -5 Can A. No; the graph fails the vertical line test. O B. Yes; there are no y-values that have more than one x-value. O C. No; there are y-values that have more than one x-value. D. Yes; the graph passes the vertical line test. ... Show moreAnswer & Explanation Solved by verified expert ec facilisis. Pellentesque dapibus efficitur laore ipiscing elit. Nam lacinia pulvinar tortor nec facilisis. Pellentesque dapibus efficitur laoreet. Nam risus ante, dapibus a molestie consequat, ultrices ac magna. Fusce dui lectus, congue vel laoreet ac, dictum vitae odio. Donec aliquet. Lorem ipsum dolor s Unlock full access to Course Hero Explore over 16 million step-by-step answers from our library Subscribe to view answer Step-by-step explanation ec facilisis. Pellentesque dapibus efficitur laoreet. Nam risus ante, dapibus a molestie consequat, ultrices ac magna. Fusce dui lectus, congue vel laoreet ac, dictum vitae odio. Donec aliquet. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nam lacinia pulvinar tortor nec facilisis. Pellentesque dapibus efficitur laoreet. Namo. Donec aliquet. Lorem ipsum dolor sit amet, conso. Donec aliquet. Lorem ipsum dolor sit amet, Learning Outcomes
As we have seen in examples above, we can represent a function using a graph. Graphs display many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis. The most common graphs name the input value [latex]x[/latex] and the output value [latex]y[/latex], and we say [latex]y[/latex] is a function of [latex]x[/latex], or [latex]y=f\left(x\right)[/latex] when the function is named [latex]f[/latex]. The graph of the function is the set of all points [latex]\left(x,y\right)[/latex] in the plane that satisfies the equation [latex]y=f\left(x\right)[/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in the graph below tell us that [latex]f\left(0\right)=2[/latex] and [latex]f\left(6\right)=1[/latex]. However, the set of all points [latex]\left(x,y\right)[/latex] satisfying [latex]y=f\left(x\right)[/latex] is a curve. The curve shown includes [latex]\left(0,2\right)[/latex] and [latex]\left(6,1\right)[/latex] because the curve passes through those points. The vertical line test can be used to determine whether a graph represents a function. A vertical line includes all points with a particular [latex]x[/latex] value. The [latex]y[/latex] value of a point where a vertical line intersects a graph represents an output for that input [latex]x[/latex] value. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because that [latex]x[/latex] value has more than one output. A function has only one output value for each input value. How To: Given a graph, use the vertical line test to determine if the graph represents a function.
Example: Applying the Vertical Line TestWhich of the graphs represent(s) a function [latex]y=f\left(x\right)?[/latex] Try ItDoes the graph below represent a function? The Horizontal Line TestOnce we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. A horizontal line includes all points with a particular [latex]y[/latex] value. The [latex]x[/latex] value of a point where a vertical line intersects a function represents the input for that output [latex]y[/latex] value. If we can draw any horizontal line that intersects a graph more than once, then the graph does not represent a function because that [latex]y[/latex] value has more than one input. How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.
Example: Applying the Horizontal Line TestConsider the functions (a), and (b)shown in the graphs below. Are either of the functions one-to-one? Identifying Basic Toolkit FunctionsIn this text we explore functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use [latex]x[/latex] as the input variable and [latex]y=f\left(x\right)[/latex] as the output variable. We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown below.
Try ItTry ItIn this exercise, you will graph the toolkit functions using an online graphing tool.
Contribute!Did you have an idea for improving this content? We’d love your input. Improve this pageLearn More Does the graph show a function explain how you know?Use the vertical line test to determine whether or not a graph represents a function. If a vertical line is moved across the graph and, at any time, touches the graph at only one point, then the graph is a function. If the vertical line touches the graph at more than one point, then the graph is not a function.
What is the range on a graph?The range is the set of possible output values, which are shown on the y -axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values.
Which of the following would be a good name for the function that takes the length of a race?The input must be the length of the race and the output must be the time since the function takes the length of the race (input) and returns the time needed to complete it (output). For a function. Therefore, Time(length) is the name of the function.
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