uc3rmw_0107e01

In this example, we are going to test for the one-to-one property of a function. We want to determine which of the following functions are one-to-one. The functions are as follows. In part A, we have f of x equal 2x plus 5. In part B, we have g of x equals x squared minus 1. And in part C, we have h of x equals 2 times the square root of x. Recall that a one-to-one function is one in which every y value corresponds to exactly one x value. We will determine whether each function is one-to-one using two methods-- the algebraic method and the geometric method-- using the horizontal line test. Let's start with the algebraic method. We can determine whether a function is one-to-one algebraically by assuming that there are two x values in the domain of the function, x1 and x2, such that f of x1 equals f of x2. Starting with f of x1 equal to f of x2, we have 2 times x1 plus 5 on the left-hand side equals 2 times x2 plus 5 on the right-hand side. We'll subtract 5 from both sides, and we have 2 times x1 equals 2 times x2. Now we just divide by 2 on both sides, and we arrive at x1 equals x2. So we can see that f of x1 equal f of x2 implies that x1 equals x2, and therefore, the function is one-to-one. Now we move on to part B. Using the same assumption-- that g of x1 is equal to g of x2-- we have x1 squared minus 1 equals x2 squared minus 1. We'll go ahead and add 1 to both sides. We have x1 squared equals x2 squared. Now we take the square root of both sides, and we have that x1 equals x2 or x1 equals minus x2. So this means that there are two distinct numbers for which g of x1 equals g of x2, and therefore, g of x is not a one-to-one function. Now we move on to part C. We apply the same assumption-- that h of x1 equals h of x2-- and we have that on the left hand side, 2 times the square root of x1. That should be equal to-- on the right-hand side-- 2 times the square root of x2. We divide by 2 on both sides, and we have the square root of x1 equal to the square root of x2. Finally, we'll take the square of both sides, and so x1 equals x2. So h of x1 equal h of x2 implies that x1 equals x2, and therefore, the function h of x is one-to-one. Now I'm going to show you the geometric method to testing for the one-to-one property of a function. In the geometric method, or the horizontal line test, we first plot the function. We then apply the horizontal line test by trying to see if there is any horizontal line we can draw that will intersect the graph of the function at more than one point. If there is no horizontal line that will pass through the graph of the function at more than one point, it is a one-to-one function. If there is a horizontal line that will pass through the graph of the function at more than one point, it is not a one-to-one function. So here we see the graph of all the functions-- f of x, g of x, and h of x. We see that no horizontal line intersects the graph of f or h in more than one point. Therefore, the functions f and h are one-to-one. The function g, on the other hand, is not one-to-one because we're able to draw a horizontal line that intersects the graph at more than one point. That line is at y equals 3, but we can see that we could draw several other horizontal lines that would intersect the graph of g at more than one point. It suffices that we only find one horizontal line that intersects the graph of g at more than one point to conclude that the function g is not one-to-one. And so that's how we're able to determine whether a function is one-to-one, both algebraically and geometrically, using the horizontal line test.