Many horizbatal lines that cut the graph of a semicircle in two points, so the horizontal line test shows that the function is not one-to-one. NOTE In Example 1(b), the graph of the function is a semicircle. HORIZONTAL LINE TEST If each horizontal line intersects the graph of a function in no more than one point, then the function is one-to-one. This horizontal line test for one-to-one functions can be summarized as follows. There is also a useful graphical test that tells whether or not a function is one-to-one. By definition, this is not a one-to-one function.Īs shown in Example 1(b), a way to show that a function is not one-to-one is to produce a pair of unequal numbers that lead to the same function value. Thus, the fact that a!=b implies that f(a)!=f(b), so f is one-to-one. ONE-TO-ONE FUNCTION A function f is a one-to-one function if, for elements aand b from the do main of f,Įxample 1 DECIDING WHETHER A FUNCTION IS ONE-TO-ONEĭecide whether each of the following functions is one-to-one. On the other hand, for the function y=x^2, two different values of x can lead to the same value of y for example, both x=4 and x=-4 give y = 4^2 = (-4)^2 =16.A function such as y = 5x -8,where different elements from the domain always lead to different elements from the range, is called a one-to-one function. ONE-TO-ONE FUNCTIONS For the function y=5x-8, any two different values of x produce two different values of y. The only functions that do have inverse functions are one-to-one functions. Not all functions have inverse functions. This section will show how to start with a function such as f(x)=8x and obtain the inverse function g(x) = (1/8)x . For these functions fand g, it can be shown that f =x and g =x for any value of x. This means that if a value of x such as x=12 is choosen, so thatĪlso, f = 12. For example, the functionsĪre inverses of each other. Similarly, some functions are inverses of each other. Notice that the nght endpoints are included in this case, instead of the left endpoints.Īddition and subtraction are inverse operations: starting with a number x, adding 5, and subtracting 5 gives x back as a result. The graph of this step function is shown in Figure 4.13. A 5.8-pound package will cost the same as a 6-pound package: 10 + 5(3) = 25, or $ 25. For a 2.5-pound package, the cost will be the same as for 3 pounds: 10 + 2(3) = 16, or $ 16. The cost for a package weighing 2 pounds is $ 10 for the first pound and $ 3 for the second pound, for a total of $ 13. Find the cost to send a package weighing 2 pounds 2.5 pounds 5.8 pounds. Each additional pound or part of a pound costs $ 3 more. The greatest integer function can be used to describe many common pricing practices encountered in everyday life, as shown in the next example.Įxample 8 APPLYING THE GREATEST INTEGER FUNCTIONĪn express mail company charges $ 10 for a package weighing 1 pound or less. NOTE The function in Example 1(b) is a good illustration of the result of combining simple functions to get a more complicated function. The graph of this quadratic function, a parabola with vertex at (2,5) opening downward, is shown in Figure 4.5. This defines a linear function with a slope of -4 and y-intercep 5. GRAPHING LINEAR AND QUADRATIC FUNCTIONS AND QUADRATIC Is a linear function, and the function defined by If a, b, and c are real numbers, then the function defined by Because of this, linear and quadratic functions can be defined as follows. By the vertical line test, any straight line that is not vertical is the graph of a function, as is the graph of any vertical parabola. Many of the graphs discussed in Chapter 3 are the graphs of functions. 4.4 - GRAPHING BASIC FUNCTIONS AND THEIR VARIATIONS
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