SECTION 1.8 : x = f LEARNING OBJECTIVES

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1 SECTION 1.8 : x = f (Section 1.8: x = f ( y) ( y)) LEARNING OBJECTIVES Know how to graph equations of the form x = f ( y). Compare these graphs with graphs of equations of the form y = f ( x). Recognize when a curve or an equation describes x as a function of y, and apply the Horizontal Line Test (HLT) for this purpose. Know basic graphs in this new context. Adapt rules and techniques for function behavior, symmetry, and transformations to this new context. PART A: DISCUSSION Sometimes, x is treated as a function of y, and we graph equations of the form x = f y ( ) in the xy-plane. These graphs must pass the Horizontal Line Test (HLT). The ordered pairs we associate with f are now of the form ( output, input). x and y switch roles. y, not x, is our independent variable, and x, not y, is our dependent variable. Function values correspond to x values. Function evaluations and point-plotting are modified accordingly. The order of the coordinates of ordered pairs must be kept in mind. We will investigate similarities and differences between graphs of equations of the form x = f ( y) and those of the familiar form y = f ( x). We will adapt tools for drawing the new graphs. For example, we will consider intervals on which f is increasing or decreasing, as well as intercepts. We will also manipulate equations to help us graph them. We will see how our rules and techniques for symmetry (from Sections 1. and 1.7) and transformations (from Section 1.4) need to be modified. We will develop a gallery of basic graphs similar to the one in Section 1.. In Section 1.9, we will see that the graphs of x = f reflections about the line y = x. ( y) and y = f ( x) are

2 PART B : GRAPHING x = f ( y) (Section 1.8: x = f ( y)) Graphing x = f ( y) As a set of ordered pairs, the graph of x = f f ( y), y ( ) {( ) y Dom f }. The graph of x = f in the usual xy-plane. ( y) is given by: ( y) is typically different from the graph of y = f ( x) When we do function evaluations and point-plotting, we treat y as the independent ( input ) variable and x as the dependent ( output ) variable. WARNING 1: We still order coordinates of points as ( x, y), but they now take the form ( output, input). PART C: SQUARING FUNCTION and EVEN FUNCTIONS Let f ( y)= y 2. f here is the same squaring function on we have always known. Its rule could also be given as: f ( x)= x 2. Dom( f )=, and Range( f )= 0, ), as before. A table for x = f Input y ( y) is below. Output x f ( y) y 2 Point x, y ( ) 0 0 0, , , 2 9 9, Input y Output x f ( y) y 2 Point x, y ( ) ( ) 0 0 ( 0, 0) ( ) 1 1 ( 1, 1) ( ) 2 4 ( 4, 2) ( ) 9 ( 9, )

3 (Section 1.8: x = f The graph of x = y 2 below is a parabola opening to the right. ( y)) 1.8. WARNING 2: The graph never goes to the left of the y-axis, because squares of real numbers are never negative. Look at the table. f is even, so each pair of opposite y values yields a common function value f ( y), or x. Graphically, this means that every point ( x, y) on the graph has a mirror image partner ( x, y) that is also on the graph, and the graph is symmetric about the x-axis (WARNING ). A function f is even f ( y)= f ( y), y Dom( f ) The graph of x = f ( y) is symmetric about the x-axis. f is decreasing on the y-interval (,0, so x values decrease there, and the graph moves to the left as we move up the graph on that interval. f is increasing on the y-interval 0, ), so x values increase there, and the graph moves to the right as we move up the graph on that interval. TIP 1: We could try to solve the equation x = f ( y) for y in terms of x. Here, x = y 2 y =± x. The graph of y = x is the top half of the parabola, while the graph of y = x is the bottom half.

4 (Section 1.8: x = f ( y)) PART D: THE HORIZONTAL LINE TEST (HLT) In Section 1.2, we said that an equation in x and y describes y as a function of x its graph passes the Vertical Line Test (VLT) in the xy-plane. We now have: The Horizontal Line Test (HLT) A curve in a coordinate plane passes the Horizontal Line Test (HLT) There is no horizontal line that intersects the curve more than once. An equation in x and y describes x as a function of y Its graph in the xy-plane passes the HLT. Then, there is no input y that yields more than one output x. Then, we can write x = f A curve could be a straight line. ( y), where f is a function. Observe that the graph of x = y 2 from Part C passes the HLT. PART E: ESTIMATING DOMAIN AND RANGE FROM A GRAPH The domain of f is the set of all y-coordinates of points on the graph of x = f ( y). (Think of projecting the graph onto the y-axis.) The range of f is the set of all x-coordinates of points on the graph of x = f ( y). (Think of projecting the graph onto the x-axis.) WARNING 4: Domain Think: y f Range Think: x If f ( y)= y 2, then, once again, Dom( f )=, and Range( f )= 0, ). The graph in Part D demonstrates this.

5 PART F: CUBING FUNCTION and ODD FUNCTIONS Let f ( y)= y. Graph x = f ( y). (Section 1.8: x = f ( y)) If we solve x = y for y, we obtain the equivalent equation y = x. We saw the graph of y = x in Section 1., and it is the graph that we want here. Observe that f is increasing on, meaning that the graph moves to the right as we move up the graph. WARNING 5: When graphing x = f ( y), trace the graph from bottom-totop (in the direction of increasing y), not top-to-bottom. A common error is to reflect the graph about the x-axis, which can happen if you rotate your head clockwise and draw the shape of the graph of y = x. f is an odd function, and we have graphical symmetry about the origin, just as before. A function f is odd f ( y)= f ( y), y Dom( f ) The graph of x = f ( y) is symmetric about the origin.

6 PART G: TRANSLATIONS ( SHIFTS ) and INTERCEPTS (Section 1.8: x = f ( y)) WARNING 6: This time, horizontal shifts are more intuitive than vertical shifts. Let G be the graph of x = f ( y). Let c be a positive real number. Horizontal Translations ( Shifts ) The graph of x = f ( y)+c is G shifted right by c units. We are increasing the x-coordinates. The graph of x = f Vertical Translations ( Shifts ) ( y) c is G shifted left by c units. The graph of x = f The graph of x = f ( y c) is G shifted up by c units. ( y+c) is G shifted down by c units. Example 1 (Translations) Let f ( y)= y 2. G is the central, purple graph of x = f y ( ) below.

7 (Section 1.8: x = f The coordinate shift method from Section 1.4, Part F still applies: Example 2 (Coordinate Shifting and Intercepts) ( y)) We will translate the parabola on the left so that its turning point (its vertex) is moved from ( 0, 0) to ( 4,1); that is, it is moved 4 units to the left and 1 unit up. ( ) 2, or ( ) 2 4 Graph of x = y 2 Graph of x + 4 = y 1 x = y 1 Look at the graph on the right. As usual, To find the x-intercept, substitute y = 0 into an equation, and solve for x. To find the y-intercepts, substitute x = 0 into an equation, and solve for y. (Finding these intercepts algebraically will be an Exercise.) When finding intercepts in the x = f ( y) setting, there are some differences. There can be at most one x-intercept, corresponding to f ( 0), if it exists. There can be any number of y-intercepts (possibly none), or infinitely many, corresponding to the real zeros of f.

8 (Section 1.8: x = f ( y)) PART H: REFLECTIONS Let G be the graph of x = f ( y). The graph of x = f ( y) is G reflected about the y-axis. (WARNING 7) The graph of x = f y The graph of x = f y ( ) is G reflected about the x-axis. ( ) is G reflected about the origin. Example (A Reflection about the y-axis) The graph of x = y 2 is in blue below. The graph of x = y 2 is in red. Example 4 (Reflections) Let f ( y)= y. G is the upper right, purple graph of x = f y Observe that x = y y = x 2 ( x 0), so G is the right half of an upward-opening parabola. ( ) below.

9 (Section 1.8: x = f ( y)) PART I: NONRIGID TRANSFORMATIONS; STRETCHING AND SQUEEZING WARNING 8: Just as for translations ( shifts ) for the x = f horizontal transformations are now the more intuitive ones. ( y) case, the ( y) is: a horizontally stretched version of G if c > 1 a horizontally squeezed version of G if 0 < c < 1 The graph of x = cf The graph of x = f ( cy) is: a vertically squeezed version of G if c > 1 a vertically stretched version of G if 0 < c < 1 If c < 0, then perform the corresponding reflection either before or after the horizontal or vertical stretching or squeezing. Example 5 (Stretching and Squeezing) Let f ( y)= y 2. Consider x = cf y ( ) on the left and x = f ( cy) on the right.

10 (Section 1.8: x = f ( y)) PART J: A GALLERY OF GRAPHS Most of the graphs below differ from those in the table in Section 1., Part P. However, x = y is equivalent to y = x, so they share a common graph. (Which other graphs are familiar from Section 1.?) Corresponding domains and ranges are the same as those in Section 1.. Again, they can be inferred from the graphs. The domain corresponds to y-coordinates this time, the range to x. Graphs of even functions are now symmetric about the x-axis. We associate increasing with move right as we move up the graphs. Equation (Sample) Graph Even/Odd; Symmetry Equation (Sample) Graph Even/Odd; Symmetry x = c Even; x-axis x = 1 y 2 Even; x-axis x = y Odd; origin x = y Neither x = cy + d ( c 0) Odd d = 0; then, origin x = y Odd; origin x = y 2 Even; x-axis x = y 2/ Even; x-axis x = y Odd; origin x = y Even; x-axis x = 1 y Odd; origin x = a 2 y 2 ( a > 0) Even; x-axis

11 (Section 1.9: Inverses of One-to-One Functions) SECTION 1.9: INVERSES OF ONE-TO-ONE FUNCTIONS LEARNING OBJECTIVES Understand the purposes and properties of inverse functions. Recognize when a function is one-to-one and invertible, and apply the Horizontal Line Test (HLT) for this purpose. Know how to find the inverse of a one-to-one function numerically and graphically, and inverse formulas conceptually and mechanically. PART A: DISCUSSION In Section 1.8, we saw that the graphs of x = y and y = x were the same. Each equation can be solved for the other variable to obtain the other equation. We call the corresponding cubing and cube root functions a pair of inverse functions. The graphs of y = x and y = x are reflections about the line y = x. The term inverse is used in many different contexts. We discussed logical inverses in Section 0.2. We will review additive and multiplicative inverses in this section. Inverse functions are inverses with respect to composition of functions (see Section 1.6). (Typically, f 1 1/ f.) Inverse properties are based on the idea that a pair of inverse functions undo each other. Inverse functions are developed by switching inputs with outputs and thus domains with ranges. In order for a function f to be invertible (to have an inverse function f 1 ), f must be one-to-one so that we obtain a function after the switch. Sometimes, the implied domain must be restricted for a function to be invertible. In Section 1.8, we used the Horizontal Line Test (HLT) to see if an equation or its graph represented x as a function of y. Now, we will use the HLT to see if a function is one-to-one and invertible. We use inverses to solve equations uniquely. For example, when solving x + 1 =, we use subtraction by 1 to invert addition by 1. In Chapter, we will see that exponential and logarithmic functions form pairs of inverse functions; in Chapter 4, we will discuss trigonometric and inverse trigonometric functions. We will solve related equations using inverse properties. In Section 8.4, we will define inverse matrices as multiplicative inverses, although they will be related to inverse functions ( transformations ) in linear algebra.

12 PART B: INVERSE FUNCTIONS (Section 1.9: Inverses of One-to-One Functions) We will begin by reviewing additive and multiplicative inverses. 0 is the additive identity of, because, if we add 0 to any real number, the result is identical to that number. Similarly, 1 is the multiplicative identity. The additive inverse of, say, the real number is. That is, is the inverse of with respect to addition. This is because, if the two numbers are added, the result is the additive identity, 0. The multiplicative inverse (or reciprocal) of is 1. That is, 1 is the inverse of with respect to multiplication. This is because, if the two numbers are multiplied, the result is the multiplicative identity, 1. A pair of inverse functions have the property that, when they are composed in either order, the result is the identity function, which outputs its input. They are inverses with respect to function composition. (See Footnote 1.) If f has an inverse function (i.e., f is invertible), then this inverse is unique, and it is denoted by f 1 (called f inverse). Also, the inverse of f 1 is f. WARNING 1: f 1, the inverse function of f, is not typically the multiplicative inverse of f. That is, typically, f 1 1 f. The 1 is just a superscript; it is not an exponent. However, ( ) f x 1 1 is interpreted as f x ( ). (See Footnote 2.) Inverse Properties (can be taken as the Definition of an Inverse Function) f has a unique inverse function f 1 f ( 1 f ( x) )= x, x Dom( f ), and f ( f 1 ( x) )= x, x Dom( f 1 ). That is, f and f 1 undo each other.

13 Example 1 (Temperature Conversion). (Section 1.9: Inverses of One-to-One Functions) 1.9. Let f be the function that converts temperature measures from the Celsius scale to the Fahrenheit scale. f 1 then exists. It is the function that converts from Fahrenheit to Celsius. For example, f ( 0)= 2 and f 1 ( 2)= 0, because 0 degrees Celsius corresponds to 2 degrees Fahrenheit. That is, 0 C = 2 F. Both measures give the freezing point of water at sea level. Also, f ( 100)= 212 and f 1 ( 212)= 100, because 100 C = 212 F. Both measures give the boiling point of water at sea level. A partial table for f and f 1 is below. Input x f Output f x ( ) Input x f 1 Output f 1 x A partial arrow diagram for f and f 1 is below. ( ) The Inverse Properties are demonstrated below. f ( 1 f ( 0) )= f 1 ( 2)= 0 f ( 1 f ( 100) )= f 1 ( 212)= 100 f ( f 1 ( 2) )= f ( 0)= 2 f ( f 1 ( 212) )= f ( 100)= 212

14 (Section 1.9: Inverses of One-to-One Functions) Example 1 demonstrates that, in going from a function to its inverse (if it exists), inputs and outputs switch roles. This is a key theme. Input-Output Properties of Inverse Functions (can also be taken as the Definition of an Inverse Function) If f has an inverse function f 1, then f ( a)= b f 1 ( b)= a. That is, ( a, b) f ( b, a) f 1. Domain and Range of Inverse Functions If f has an inverse function f 1, then: Dom( f )= Range( f 1 ), and Dom( f 1 )= Range( f ). That is, the domain of one function is the range of the other. Example 2 (Inverse Functions: Verification, Domain, and Range) a) Let f ( x)= x on. Find f 1, and verify that it is the inverse of f. b) Let g( x)= x on 0, 2. Find g 1. c) Let h( x)= x on { 2,1, }. Find h 1. Solution The inverse of each cubing function above is a cube root function. a) f 1 ( x)= x on, which is Range( f ). Let r( x)= x on. We will verify that r is the inverse of f by verifying the Inverse Properties for f and r. ( r f )( x)= r( f ( x) )= r( x )= x = x, x, and ( f r) ( x)= f ( r( x) )= f ( x)= ( x ) = x, x. b) g 1 ( x)= x on 0, 8, which is Range( g). c) h 1 ( x)= x on { 8, 1, }, which is Range( h). The reader can investigate further in the Exercises.

15 PART C: ONE-TO-ONE FUNCTIONS and THE HORIZONTAL LINE TEST (HLT) We now discuss which functions are invertible. From Section 1.1, (Section 1.9: Inverses of One-to-One Functions) A relation f is a function Each input to f in its domain yields exactly one output in its range. ( ) in the xy-plane passes the Vertical Line Test (VLT). The graph of y = f x A function cannot allow an input in its domain to yield two or more different outputs. The following do not represent functions: Input-Output Machine Arrow Diagram Graph and Equation Fails VLT Now, y =± 9 x 2 A function f is one-to-one Each output from f in its range is yielded by exactly one input in its domain. Equivalently, f ( a)= f ( c) a = c. That is, identical outputs imply identical inputs. (See Footnote.) A one-to-one function cannot allow two or more different inputs (x values) in its domain to yield the same output (y value).

16 (Section 1.9: Inverses of One-to-One Functions) Example (A Function that is Not One-to-One: Unrestricted Squaring Function) Let f ( x)= x 2 on. Then, f is a function, but it is not one-to-one on, as demonstrated by the figures below: Input-Output Machine Partial Arrow Diagram Graph Fails HLT (Section 1.8) Also, f ( a)= f c a = c, if c 0. ( ) a 2 = c 2 a =±c, which is not equivalent to If we solve the equation f ( x)= 9, or x 2 = 9, we obtain two solutions for x, namely and. These are the two answers to the question, Whose square is 9? When we switch inputs with outputs, the expression f 1 ( 9) is not well-defined here, because there are two possible values it could take on: and. As a consequence, f is not invertible, because it has no inverse function. However, a squaring function can be one-to-one and invertible on a restricted domain, as we will see in Example 4. We will restrict domains when we define inverse trigonometric functions in Section 4.10.

17 (Section 1.9: Inverses of One-to-One Functions) Example 4 (A One-to-One Function: Squaring Function on a Restricted Domain; Modifying Example ) Let g( x)= x 2 on the restricted domain 0,). The graph of y = g( x) below passes the Vertical Line Test (VLT) and also the Horizontal Line Test (HLT). Consequently, g is a one-to-one correspondence between Dom( g), the set of input x values, and Range( g), the set of output y values. (Think of matched pairs.) Also, g( a)= g( c) a 2 = c 2 a = c, since only nonnegative inputs are allowed. If we solve the equation g( x)= 9, or x 2 = 9 ( x 0), we obtain a unique solution for x, namely. It is the unique input that yields 9. On the graph of g above, the only point with y-coordinate 9 has x-coordinate. More generally, g( x)= b has a unique solution for x, the unique input that yields b, whenever b is in Range( g), which is 0,). We can define a unique inverse function g 1. Let Dom( g 1 )= Range( g), which is 0,). Define g 1 ( b) to be the unique solution to g( x)= b, for every b in Dom( g 1 ). For instance, g 1 ( 9)=. (In Example 1, since f ( 0)= 2, we reverse the arrow and define f 1 2 ( ) to be 0.)

18 (Section 1.9: Inverses of One-to-One Functions) In summary A function f is invertible f ( x)= b has a unique solution, given by x = f 1 ( b), b Range f f is one-to-one The graph of y = f x ( ) ( ) in the xy-plane passes the Horizontal Line Test (HLT). The one-to-one property is essential for a function to be invertible, because we need the inverse to be a function after inputs are switched with outputs. (See Footnote ; we assume the onto property.) PART D: GRAPHING INVERSE FUNCTIONS By the Input-Output Properties, ( a, b) f ( b, a) f 1. Graphical Properties of Inverse Functions If f is invertible, then: The point a,b The point b,a ( ) lies on the graph of f ( ) lies on the graph of f 1. To obtain the graph of f 1, reflect the graph of f about the line y = x. Example 5 (Restricted Squaring Function; Revisiting Example 4) Again, let g( x)= x 2 on 0,). Partial tables for g and g 1 can be constructed as follows: x g g x g 1 ( ) Point x g 1 ( x) Point ( ) 0 0 ( 0, 0) ( ) 1 1 ( 1, 1) ( ) 4 2 ( 4, 2) ( ) 9 ( 9, ) 0 0 0, , , 4 9, 9

19 (Section 1.9: Inverses of One-to-One Functions) The graph of g, or y = g( x), is in blue below. It passes the HLT, meaning that g is one-to-one. Therefore, when we reflect the graph about the line y = x (drawn as a dashed line, though it is not an asymptote), we obtain the red graph of y = g 1 x ( ) below, and it passes the VLT. It should look familiar.. The graph of y = g 1 ( x) is the graph of y = x, because g 1 ( x)= x. It makes sense that the corresponding square root function undoes what the (restricted) squaring function does. It is also the graph of x = g( y), or x = y 2, with y restricted to 0,). This is consistent with the aforementioned Graphical Properties of Inverse Functions. ( ) for a function g is the reflection of the graph ( ) about the line y = x. The graph of x = g y of y = g x This holds even if g is not one-to-one.

20 (Section 1.9: Inverses of One-to-One Functions) PART E: FINDING FORMULAS FOR INVERSE FUNCTIONS If f is a one-to-one function, and if its formula can be expressed algebraically, then we should be able to find a formula for f 1. Conceptual Approach to Finding the Inverse of a One-to-One Function f To determine f 1, we need to invert (or undo ) the steps applied by f in reverse order (WARNING 2). (See the Exercises.) Also, if necessary, impose restrictions and ensure that Dom( f )= Range( f 1 ), and Dom( f 1 )= Range( f ). (See Example 10.) TIP 1: Similarly, when dressing, you put on your socks before your shoes, but, when undressing, you remove your shoes before your socks. Example 6 (Conceptual Approach to Finding an Inverse) Solution Let f ( x)= x on. Find f 1 ( x). What does f do to x? 1) Takes the cube root: input 2) Adds 5 to the result from 1): input +5 ) Divides the result from 2) by 7: input 7 What should f 1 do to its input? (The asterisk * denotes inverting.) *) Multiplies by 7: input 7 2*) Subtracts 5 from the result from *): input 5 ( ) 1*) Cubes the result from 2*): input Therefore, f 1 ( x)= 7x 5 ( ). Dom( f )= Range( f 1 )=, and Dom( f 1 )= Range( f )=, so no restrictions are necessary.

21 (Section 1.9: Inverses of One-to-One Functions) Mechanical Approach to Finding the Inverse of a One-to-One Function f Given a formula for f in terms of x, we can attempt to find a formula for f 1 as follows: Step 1: Replace f ( x) with y. Step 2: Switch x and y. (Remember the theme of switching inputs with outputs.) Step : Solve for y, if possible. Step 4: Replace y with f 1 ( x). Step 5: If necessary, impose restrictions and ensure that Dom( f )= Range( f 1 ), and Dom( f 1 )= Range( f ). In Steps 1-, we are essentially solving x = f ( y) for y. Example 7 (Mechanical Approach to Finding an Inverse; Revisiting Example 6) Again, let f ( x)= x on. Find f 1 ( x). Solution Step 1: Replace f ( x) with y. y = x Step 2: Switch x and y. (Remember the theme of switching inputs with outputs.) x = y + 5 7

22 Step : Solve for y. (Section 1.9: Inverses of One-to-One Functions) x = 7x 5 = ( ) ( ) ( ) y + 5 Multiplied both sides by 7 y Subtracted 5 from both sides ( 7x 5) = y Cubed both sides WARNING : Squaring both sides of an equation may or may not yield an equivalent equation. y = ( 7x 5) Compare these steps with Steps *, 2*, and 1* in Example 6. Step 4: Replace y with f 1 ( x), if we obtain a function. f 1 ( x)= 7x 5 ( ) f 1 is a function, because f was one-to-one. Step 5: If necessary, impose restrictions and ensure that Dom( f )= Range( f 1 ), and Dom( f 1 )= Range( f ). Dom( f )= Range( f 1 )=, and Dom( f 1 )= Range( f )=, so no restrictions are necessary. Example 8 (Evaluating an Inverse Function; Revisiting Examples 6 and 7) Solution Again, let f ( x)= x f 1 ( x)= 7x 5 f 1 1 on. Evaluate f 1 (). 1 ( ) on ( from Examples 6 and 7) ( ) ()= 71 ()5 = 8 Absent Examples 6 and 7, we could have also solved f ( x)= 1 for x.

23 (Section 1.9: Inverses of One-to-One Functions) Example 9 (Checking an Inverse Function Formula; Revisiting Examples 6 and 7) Again, let f ( x)= x Check that g = f 1. on. Let g( x)= ( 7x 5) on. Solution We will verify that the compositions g f and f g are identity functions. Either check below is sufficient, because f is one-to-one, and Range( f )= Dom( g) (each set is ). If in doubt, do both checks. (See Footnote 1.) Check that ( g f )( x)= x, x. Check that ( f g) ( x)= x, x. x, x, ( g f ) x ( ( )) ( )= g f x = g = 7 x x = x = x = x Therefore, g = f 1. ( f g) x ( ( )) ( ) ( )= f g x ( ) = f 7x 5 ( 7x 5) = x = 7 = 7x 7 = x

24 (Section 1.9: Inverses of One-to-One Functions) Example 10 (Temperature Conversion; Revisiting Example 1) Let f be the one-to-one function that converts from Celsius to Fahrenheit. Then, f 1 is the function that converts from Fahrenheit to Celsius. The following is left for the reader in the Exercises: Show that f ( x)= 9 x + 2 by developing a linear model for f such 5 that f ( 0)= 2 and f ( 100)= 212. Show that f 1 ( x)= 5 9 ( x 2) in three different ways: Develop a linear model for f 1 such that f 1 ( 2)= 0 and f 1 ( 212)= 100. Begin with f ( x)= 9 x + 2 and apply the Conceptual 5 Approach used in Example 6. Begin with f ( x)= 9 x + 2 and apply the Mechanical 5 Approach used in Example 7. We will now determine domains and ranges. (See Step 5 in the Mechanical Approach.) Temperatures cannot go below absolute zero, which is C. We require: Dom( f )= Range( f 1 )= { x x 27.15}. Find the Fahrenheit equivalent of C. f ( x)= 9 x f ( 27.15)= 9 ( 27.15)+ 2 5 = F ( ) We require: Dom( f 1 )= Range( f )= { x x }.

25 (Section 1.9: Inverses of One-to-One Functions) Observe that the red graph for f and the brown graph for f 1 below are reflections about the line y = x. FOOTNOTES 1. Identity functions and compositions of inverse functions. There are technically different identity functions on different domains. (See Footnote below and Section 1.1, Footnote 1.) Let f be an invertible function that maps from domain X to codomain Y ; i.e., f : X Y. If f is invertible, then f is onto, meaning that the range of f is the codomain Y. f 1 maps from Y to X ; i.e., f 1 :Y X. Let I X be the identity function on Dom( f ), which is X. I X : X X. Let I Y be the identity function on Dom( f 1 ), which is Y. I Y :Y Y. Then, f 1 f = I X, and f f 1 = I Y. If g is a function such that g f = I X, then g is a left inverse of f ; f has a left inverse f is one-to-one. For example, let X = { 1, 2}, Y = { 10, 20, 0}, f = {( 1, 10), ( 2,20) }, and g = {( 10,1), ( 20,2), ( 0,2) }. Then, g is a left inverse of f. If h is a function such that f h = I Y, then h is a right inverse of f ; f has a right inverse f is onto. For example, let X = { 1, 2, }, Y = { 10, 20}, f = {( 1, 10), ( 2,20), (,20) }, and h = {( 10, 1), ( 20,2) }. Then, h is a right inverse of f, although h is not a left inverse of f. If f is one-to-one and onto, then f has a unique inverse function that serves as both a unique left inverse and a unique right inverse.

26 (Section 1.9: Inverses of One-to-One Functions) f n. Many instructors reluctantly use the f 1 notation to represent the inverse function of f. This is because n often represents an exponent in the notation f n, except when n = 1. For example, f 2 is often taken to mean ff ; that is, f 2 ( x)= f ( x) f ( x). In Chapters 4 and 5, we will accept that sin 2 x = ( sin x) ( sin x), which is the standard interpretation. On the other hand (and this compounds the confusion), some sources use n to indicate the number of applications of f in compositions of f with itself; the result is called an iterated function. For example, they would let f 2 = f f, and they would use the rule: f 2 ( x)= f ( f ( x) ). This is typically different from the rule f 2 ( x)= f ( x) f ( x). However, our use of the notation f 1 for f inverse is more consistent with this second interpretation, since f 1 f is an identity function, which could be construed as f 0 in this context.. One-to-one, onto, and bijective functions. A function f is invertible it is one-to-one (or injective) and onto (or surjective); then, f is a bijective function, or a one-to-one correspondence. A one-to-one (or injective ) function has the property that, whenever f ( a)= f ( c) for domain elements a and c, it must be true that a = c. That is, two outputs are equal the inputs are equal. This definition will lead to the One-to-One Properties for exponential and logarithmic functions in Chapter. An onto (or surjective ) function has the property that its range is the entire codomain (see Section 1.1, Footnote 1). This means that every element of the codomain is the image (that is, function value or output) of some element of the domain. When we develop the inverse of a bijective function, we switch inputs with outputs. The one-to-one property guarantees that the resulting function does not allow one input to yield more than one output. The onto property guarantees that the resulting function is defined for all elements of the codomain of the original function, which is now the domain of the inverse function. The onto property is often ignored in discussions of inverse functions. This is because we typically force the codomain of the original function to be the range (or image ) of the original function in this context; this then guarantees the onto property. (Precalculus sources typically avoid the term codomain in the first place.) If this is the case, then one-to-one function and one-to-one correspondence are interchangeable, and we only have to worry about the one-to-one property when it comes to invertibility.