When you’re asked to draw a function and its inverse, you may choose to draw this line in as a
dotted line; this way, it acts like a big mirror, and you can literally see the points of the function reflecting over the line to become the inverse function points. Reflecting over that line switches the x and the y and gives you a graphical way to find the inverse without plotting tons of points. The best way to understand this concept is to see it in action. For instance, say that you know these two functions are inverses of each other: To see how x and y switch places, follow these steps:
The entire domain and range swap places from a function to its inverse. For instance, knowing that just a few points from the given function f(x) = 2x – 3 include (–4, –11), (–2, –7), and (0, –3), you automatically know that the points on the inverse g(x) will be (–11, –4), (–7, –2), and (–3, 0). So if you’re asked to graph a function and its inverse, all you have to do is graph the function and then switch all x and y values in each point to graph the inverse. Just look at all those values switching places from the f(x) function to its inverse g(x) (and back again), reflected over the line y = x.You can now graph the function f(x)= 3x – 2 and its inverse without even knowing what its inverse is. Because the given function is a linear function, you can graph it by using the slope-intercept form. First, graph y = x. The slope-intercept form gives you the y-intercept at (0, –2). Since the slope is 3=3/1, you move up 3 units and over 1 unit to arrive at the point (1, 1). If you move again up 3 units and over 1 unit, you get the point (2, 4). The inverse function, therefore, moves through (–2, 0), (1, 1), and (4, 2). Both the function and its inverse are shown here. About This ArticleThis article is from the book:
About the book author:Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. This article can be found in the category:
Contents: This page corresponds to § 1.7 (p. 150) of the text. Suggested Problems from Text p.158 #1-4, 5, 8, 9, 12, 13, 15, 18, 21, 22, 27, 31, 34, 37, 46, 48, 51, 71, 74, 83
Definition of Inverse FunctionBefore defining the inverse of a function we need to have the right mental image of function. Consider the function f(x) = 2x + 1. We know how to evaluate f at 3, f(3) = 2*3 + 1 = 7. In this section it helps to think of f as transforming a 3 into a 7, and f transforms a 5 into an 11, etc. Now that we think of f as "acting on" numbers and transforming them, we can define the inverse of f as the function that "undoes" what f did. In other words, the inverse of f needs to take 7 back to 3, and take -3 back to -2, etc. Let g(x) = (x - 1)/2. Then g(7) = 3, g(-3) = -2, and g(11) = 5, so g seems to be undoing what f did, at least for these three values. To prove that g is the inverse of f we must show that this is true for any value of x in the domain of f. In other words, g must take f(x) back to x for all values of x in the domain of f. So, g(f(x)) = x must hold for all x in the domain of f. The way to check this condition is to see that the formula for g(f(x)) simplifies to x. g(f(x)) = g(2x + 1) = (2x + 1 -1)/2 = 2x/2 = x. This simplification shows that if we choose any number and let f act it, then applying g to the result recovers our original number. We also need to see that this process works in reverse, or that f also undoes what g does. f(g(x)) = f((x - 1)/2) = 2(x - 1)/2 + 1 = x - 1 + 1 = x. Letting f-1 denote the inverse of f, we have just shown that g = f-1. Definition:
Exercise 1:
Return to Contents Graphs of Inverse FunctionsWe have seen examples of reflections in the plane. The reflection of a point (a,b) about the x-axis is (a,-b), and the reflection of (a,b) about the y-axis is (-a,b). Now we want to reflect about the line y = x.
Let f(x) = x3 + 2. Then f(2) = 10 and the point (2,10) is on the graph of f. The inverse of f must take 10 back to 2, i.e. f-1(10)=2, so the point (10,2) is on the graph of f-1. The point (10,2) is the reflection in the line y = x of the point (2,10). The same argument can be made for all points on the graphs of f and f-1. The graph of f-1 is the reflection about the line y = x of the graph of f.
Return to Contents Existence of an InverseSome functions do not have inverse functions. For example, consider f(x) = x2. There are two numbers that f takes to 4, f(2) = 4 and f(-2) = 4. If f had an inverse, then the fact that f(2) = 4 would imply that the inverse of f takes 4 back to 2. On the other hand, since f(-2) = 4, the inverse of f would have to take 4 to -2. Therefore, there is no function that is the inverse of f. Look at the same problem in terms of graphs. If f had an inverse, then its graph would be the reflection of the graph of f about the line y = x. The graph of f and its reflection about y = x are drawn below. Note that the reflected graph does not pass the vertical line test, so it is not the graph of a function. This generalizes as follows: A function f has an inverse if and only if when its graph is reflected about the line y = x, the result is the graph of a function (passes the vertical line test). But this can be simplified. We can tell before we reflect the graph whether or not any vertical line will intersect more than once by looking at how horizontal lines intersect the original graph! Horizontal Line Test
The property of having an inverse is very important in mathematics, and it has a name. Definition: A function f is one-to-one if and only if f has an inverse. The following definition is equivalent, and it is the one most commonly given for one-to-one. Alternate Definition: A function f is one-to-one if, for every a and b in its domain, f(a) = f(b) implies a = b.
Return to Contents Finding InversesExample 1. First consider a simple example f(x) = 3x + 2.
Steps for finding the inverse of a function f.
Step 2 often confuses students. We could omit step 2, and solve for x instead of y, but then we would end up with a formula in y instead of x. The formula would be the same, but the variable would be different. To avoid this we simply interchange the roles of x and y before we solve. Example 3. f(x) = x3 + 2
Exercise 3:
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