One-to-One Functions

1. One-to-One Functions To show that these two sets are related, G is called the inverse of F. For a function  to have an inverse, must be a one-to-one function. In a one-to-one function, each x-value corresponds to only one y-value, and each y-value corresponds to only one x-value.

2. One-to-One Functions This function is not one-to-one because the y-value 7 corresponds to two x-values, 2 and 3. That is, the ordered pairs (2, 7) and (3, 7) both belong to the function.

3. One-to-One Functions This function one-to-one.

4. One-to-One Function A function is a one-to-one function if, for elements a and b in the domain of ,

5. Horizontal Line Test As shown in Example 1(b), a way to show that a function is not one-to-one is to produce a pair of different numbers that lead to the same function value. There is also a useful graphical test, the horizontal line test, that tells whether or not a function is one-to-one.

6. Horizontal Line Test If any horizontal line intersects the graph of a function in no more than one point, then the function is one-to-one.

7. Tests to Determine Whether a Function is One-to-One 1. Show that (a) = (b) implies a = b. This means that is one-to-one. (Example 1(a)) 2. In a one-to-one function every y-value corresponds to no more than one x-value. To show that a function is not one-to-one, find at least two x-values that produce the same y-value. (Example 1(b))

8. Tests to Determine Whether a Function is One-to-One 3. Sketch the graph and use the horizontal line test. (Example 2) 4. If the function either increases or decreases on its entire domain, then it is one-to-one. A sketch is helpful here, too. (Example 2(b))

9. Inverse Function Let be a one-to-one function. Thengis the inverse function of  if for every x in the domain of g, and for every x in the domain of .

10. DECIDING WHETHER TWO FUNCTIONS ARE INVERSES Example 3 Let functions andgbe defined by and , respectively. Is g the inverse function of ? Solution The horizontal line test applied to the graph indicates that is one-to-one, so the function does have an inverse. Since it is one-to-one, we now find ( ◦ g)(x) and (g ◦ )(x).

11. DECIDING WHETHER TWO FUNCTIONS ARE INVERSES Example 3 Let functions andgbe defined by and , respectively. Is g the inverse function of ? Solution

12. DECIDING WHETHER TWO FUNCTIONS ARE INVERSES Example 3 Let functions andgbe defined by and , respectively. Is g the inverse function of ? Solution Since ( ◦ g)(x) = xand (g ◦ )(x) = x, function gis the inverse of function .

13. Special Notation A special notation is used for inverse functions: If gis the inverse of a function , thengis written as -1 (read “-inverse”). For (x) = x3– 1,

14. Caution Do not confuse the –1in -1 with a negative exponent. The symbol -1(x) does not represent ;it represents the inverse function of .

15. Inverse Function By the definition of inverse function, the domain of is the range of -1, and the range of  is the domain of -1 .

16. Finding the Equation of the Inverse of y = (x) For a one-to-one function defined by an equation y = (x), find the defining equation of the inverse as follows. (You may need to replace (x) with y first.) Step 1 Interchange x and y. Step 2 Solve for y. Step 3 Replace y with -1(x).

17. FINDING EQUATIONS OF INVERSES Example 5 Decide whether each equation defines a one-to-one function. If so, find the equation of the inverse. a. Solution The graph of y = 2x + 5 is a non-horizontal line, so by the horizontal line test, is a one-to-one function. To find the equation of the inverse, follow the steps in the preceding box, first replacing (x) with y.

18. FINDING EQUATIONS OF INVERSES Example 5 Solution y = (x) Interchange x and y. Solve for y. Replace y with -1(x).

19. GRAPHING THE INVERSE Example 6 Solution

20. FINDING THE INVERSE OF A FUNCTION WITH A RESTRICTED DOMAIN Example 7 Find Let SolutionFirst, notice that the domain of is restricted to the interval [–5, ). Function is one-to-one because it is increasing on its entire domain and, thus, has an inverse function. Now we find the equation of the inverse.

21. FINDING THE INVERSE OF A FUNCTION WITH A RESTRICTED DOMAIN Example 7 Solution y = (x) Interchange x and y. Square both sides. Solve for y.

22. FINDING THE INVERSE OF A FUNCTION WITH A RESTRICTED DOMAIN Example 7 Solution However, we cannot define -1 as x2– 5. The domain of is [–5, ), and its range is [0, ).The range of is the domain of -1, so -1 must be defined as

23. FINDING THE INVERSE OF A FUNCTION WITH A RESTRICTED DOMAIN Example 7 As a check, the range of -1, [–5, ), is the domain of . Graphs of  and -1 are shown. The line y = x is included on the graphs to show that the graphs are mirror images with respect to this line.

24. Important Facts About Inverses 1. If is one-to-one, then -1 exists. 2. The domain of is the range of -1,and the range of  is the domain of -1. 3. If the point (a, b) lies on the graph of , then (b, a) lies on the graph of -1, so the graphs of and -1 are reflections of each other across the line y = x. 4. To find the equation for -1, replace (x) with y, interchange x and y, and solve for y. This gives -1 (x).