It seems like all equations would be considered functions. Take a look at an example that is not considered a function. When we input 4 for x, we must take the square root of both sides in order to solve for y. Therefore, this does not satisfy the definition for a function: "the value of the first variable corresponds to one and only one value for the second value".
We have more than one value for y. Hopefully with these two examples, you now understand the difference between an equation that represents a function and an equation that does not represent a function. You will find more examples as you study the lessons in this chapter.
So, what kinds of functions will you study? In Algebra 1, we will study linear functions much like linear equations and quadratic functions. As you progress into Algebra 2, you will be studying exponential functions. Below is the table of contents for the Functions Unit.
Click on the lesson that interests you, or follow them in order for a complete study of functions in Algebra 1. Click here for more information on our affordable subscription options. Multiplying and dividing functions Opens a modal. Composing functions Algebra 2 level. Intro to composing functions Opens a modal. Composing functions Opens a modal. Evaluating composite functions Opens a modal. Evaluating composite functions: using tables Opens a modal.
Evaluating composite functions: using graphs Opens a modal. Finding composite functions Opens a modal. Evaluating composite functions advanced Opens a modal. Evaluate composite functions. Find composite functions. Shifting functions. Shifting functions examples Opens a modal. Graphing shifted functions Opens a modal. Shift functions. Stretching functions. Identifying function transformations Opens a modal. Identifying horizontal squash from graph Opens a modal.
Identify function transformations. Modeling situations by combining and composing functions Algebra 2 level. Modeling with function combination Opens a modal. Modeling with composite functions Opens a modal. Modeling with composite functions: skydiving Opens a modal. Model with function combination. Model with composite functions. Introduction to inverses of functions Algebra 2 level.
Intro to inverse functions Opens a modal. Graphing the inverse of a linear function Opens a modal. Evaluate inverse functions. Finding inverse functions Algebra 2 level. Finding inverse functions: linear Opens a modal. Finding inverse functions: quadratic Opens a modal. Finding inverse functions: quadratic example 2 Opens a modal.
Finding inverse functions: radical Opens a modal. Finding inverses of rational functions Opens a modal. Finding inverse functions Opens a modal. Finding inverses of linear functions. Verifying that functions are inverses Algebra 2 level. Verifying inverse functions by composition Opens a modal.
Verifying inverse functions by composition: not inverse Opens a modal. We talked briefly about this when we gave the definition of the function and we saw an example of this when we were evaluating functions.
We now need to look at this in a little more detail. Note that we did mean to use equation in the definitions above instead of functions. These are really definitions for equations. However, since functions are also equations we can use the definitions for functions as well.
We are much more interested here in determining the domains of functions. At this point, that means that we need to avoid division by zero and taking square roots of negative numbers. If we remember these two ideas finding the domains will be pretty easy. The domain is then,.
This one is going to work a little differently from the previous part. In this case it will be just as easy to directly get the domain. To avoid square roots of negative numbers all that we need to do is require that.
So, all we need to do then is worry about the square root in the numerator. Therefore, the domain of this function is. So, to keep the square root happy i. Again, to do this simply set the denominator equal to zero and solve. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode.
If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.
Example 1 The following relation is a function. Therefore, this relation is a function. Example 2 The following relation is not a function.
Example 3 Determine which of the following equations are functions and which are not functions. So, this equation is a function. So, it seems like this equation is also a function. Here are the evaluations for this part. The graph for this function is below.
The degree of the function is 3, therefore it is a cubic function. In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function, or not. If all vertical lines intersect a curve at most once then the curve represents a function. Vertical Line Test: Note that in the top graph, a single vertical line drawn where the red dots are plotted would intersect the curve 3 times.
Thus, it fails the vertical line test and does not represent a function. Any vertical line in the bottom graph passes through only once and hence passes the vertical line test, and thus represents a function.
If, alternatively, a vertical line intersects the graph no more than once, no matter where the vertical line is placed, then the graph is the graph of a function. For example, a curve which is any straight line other than a vertical line will be the graph of a function. Apply the vertical line test to determine which graphs represent functions.
Applying the Vertical Line Test: Which graphs represent functions? If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. From this we can conclude that these two graphs represent functions. This is shown in the diagram below. Not a Function: The vertical line test demonstrates that a circle is not a function. Privacy Policy.
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