Reciprocal Graphs

6 Key excerpts on "Reciprocal Graphs"

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  eBook - PDF

  College Algebra

  • Sheldon Axler(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  The Graph of an Inverse Function We begin with an example that illustrates how the graph of an inverse function is related to the graph of the original function. example 1 Suppose f is the function with domain [0, 2] defined by f (x) = x 2 . What is the relationship between the graph of f and the graph of f -1 ? solution The graph of f is part of the familiar parabola defined by the curve y = x 2 . The range of f is the interval [0, 4]. The inverse function f -1 has domain [0, 4], with f -1 (x) = √ x. The graphs of f and f -1 are shown below. These graphs are symmetric about the line y = x, meaning that we could obtain either graph by flipping the other graph across this line. 1 2 3 4 x 1 2 3 4 y The graph of x 2 (blue) and the graph of its inverse √ x (red) are symmetric about the line y = x (black). The relationship noted above between the graph of x 2 and the graph of 1 2 x 1 2 y Flipping the point (2, 1) (blue) across the line y = x gives the point (1, 2) (red). its inverse √ x holds in general for the graph of any one-to-one function and the graph of its inverse. Suppose, for example, that the point (2, 1) is on the graph of some one-to-one function f . This means that f (2) = 1, which is equivalent to the equation f -1 (1) = 2, which means that (1, 2) is on the graph of f -1 . The figure in the margin shows that the point (1, 2) can be obtained by flipping the point (2, 1) across the line y = x. 198 chapter 3 Functions and Their Graphs More generally, a point (a, b) is on the graph of a one-to-one function f if and only if (b, a) is on the graph of its inverse function f -1 . In other words, the graph of f -1 can be obtained by interchanging the first and second coordinates of each point on the graph of f . Interchanging first and second coordinates amounts to flipping across the line through the origin with slope 1 (which is the line y = x if we are working in the xy -plane).

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  eBook - PDF

  N1 Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  I = V R is an example of an inverse relationship. I R I = V R 174 Module 5 • Algebraic graphs Activity 5.7 1. Study the tables and graphs below. Indicate whether the x and y -values are directly proportional or inverse proportional. a) x 2 4 8 10 12 20 y 10 20 40 50 60 100 b) x 1 2 4 5 10 20 y 20 10 5 4 2 1 c) y x f ( x ) d) y x g ( x ) e) y x h ( x ) f) y x i ( x ) 2. Use the equation y = – 5 x to answer the questions. a) What type of graph is y = – 5 x ? b) In which quadrant(s) would you draw the graph? c) Give the y -intercept of the graph. d) Is this a directly or an inverse proportional graph? 3. Given: y = 6 x and y = – 1 2 x + 3. a) Which one is a straight-line graph? b) Is the gradient of the straight line positive or negative? c) What is the gradient of the straight line? d) Give the name of the other graph. e) In which quadrant(s) would the other graph be? f) Give the y -intercept of the straight-line graph. g) Write down the equation of the directly proportional graph. h) Write down the equation of the inverse proportional graph. 175 N1 Mathematics| Hands-On! Summary of module 5: Algebraic graphs Before you do the summative assessment, you should know the following. • Representing ordered number pairs (coordinates) on a Cartesian plane : ( x ; y ) y x y x O ( x ; y ) (– x ; – y ) • Function notation: Write a linear equation of the form y = mx + c in the form f ( x ) = mx + c and a rectangular hyperbola of the form y = c x in the form f ( x ) = c x . A function is a relationship between two variables x and y, which means that for every x -value ( independent variable ) there is only one y -value ( dependent variable ). No two coordinates have the same x -values. If y = –2 x + 1 If y = – 2 x ∴ f ( x ) = –2 x + 1 ∴ f ( x ) = – 2 x • How to draw a graph » Decide on a proper scale for both axes. Use the highest and lowest x -values in the table to find the domain of the graph. Use the highest and lowest y -values in the table to find the range of the graph.

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  eBook - PDF

  Mathematics N4 Student's Book

  TVET FIRST

  • SA Chuturgoon(Author)
  • 2022(Publication Date)
  • Macmillan

    (Publisher)

  113 Functions and graphs TVET FIRST Module 4 Inverse functions produce graphs that are reflections over the line y = x. These inverse functions have reversed ordered pairs, meaning that the domain of one graph becomes the range of the other graph and vice versa. (See Example 4.14 in section 4.2.8.) The inverse function of x is denoted as: f −1 (x) Figure 4.5: Notation for inverse functions Example 4.3 Find the inverse function of: 1. y = 3x + 2 2. y = log a x 3. y = ln x Solution 1. y = 3x + 2 [Also f (x) = 3x + 2] x = 3y + 2 [Interchange x and y] 3y + 2 = x [Make y the subject of the equation] 3y = x − 2 y = x − 2 _ 3 ∴ f −1 (x) = x − 2 _ 3 2. y = log a x [Also f (x) = log a x] x = log a y log a y = x y = a x ∴ f −1 (x) = a x 3. y = ln x [Also f (x) = ln x] y = log e x x = log e y log e y = x y = e x ∴ f −1 (x) = e x 4.1.5 Continuous and discontinuous graphs Continuous functions and relations If a function or relation produces a graph that can be drawn completely without lifting the pencil, then this is a continuous function or relation. There is no interruption or break in the graph at any point. An equation will have an inverse function if a line segment drawn parallel to the x-axis cuts the graph once only. Important continuous: without interruption reflection: the image when a graph of a function is mirrored (flipped) about an axis or line 114 Module 4 TVET FIRST y x y x y x Figure 4.6: Examples of continuous graphs Discontinuous functions and relations If a function or relation produces a graph that cannot be drawn completely without lifting the pencil, then this function or relation is discontinuous. y x y x x y Figure 4.7: Examples of discontinuous graphs 4.1.6 Symmetry Definition of axis of symmetry The axis of symmetry is a straight line that divides a graph into two identical images or produces two mirror images. Figure 4.8 illustrates some examples of symmetrical functions and relations.

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  eBook - PDF

  Teaching and Learning Algebra

  • Doug French(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  When the graph is extended to positive real numbers, the behaviour as x approaches zero offers similar insights, together with a link to the idea of reciprocal and its self-inverse properties. At a later stage, the implications of the equivalent form xy — 1 can be explored with its links to inverse proportion, discussed in Chapter 8. 96 Teaching and Learning Algebra The form y = —-+ q for a translation of y = ^ through/? units in the x direction and q units in the y direction is no different to that for other functions, but its effect in moving the vertical asymptote to x = p and the horizontal asymptote is y = q is an important con-sideration. As with y = jc 2 , it is, for most purposes, better to consider stretches in the y direction only when interpreting the family of curves given by y = ^, as displayed in Figure 6.19. Students tend to see the stretch as being in the direction indicated by the line y -x 9 but this is not so convenient as a stretch in either the x or y direction where the scale factor is simply c. Figure 6.19 The family of curves y «• * for x > 0 Investigating the graph of the rational function y = ^~f by looking at the effect of varying the values of the constants a and b provides students a valuable opportunity to develop skills with algebraic fractions and graphical interpretation. Figure 6.20 shows two functions of the form y = ^5f > b ot h w ith 2 as the value of b. If students are asked what they notice about these and other curves generated by the same function, they will comment on the intersections with the axes and the asymptotes. In Chapter 2 I referred to the notion of a 'procept', put forward by Gray and Tall (1993), to bring together the ideas of process and product. This relates well to ~5f» which students initially see as representing a 'process' in the sense that it enables numbers to be calculated to determine points on a graph and can therefore serve as a label for a graph.

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  eBook - PDF

  Precalculus with Limits

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  90 Chapter 1 Functions and Their Graphs 1.9 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. GO DIGITAL Vocabulary and Concept Check In Exercises 1–4, fill in the blanks. 1. If f (g(x)) and g( f (x)) both equal x, then the function g is the ________ function of the function f. 2. The inverse function of f is denoted by ________. 3. The domain of f is the ________ of f -1 , and the ________ of f -1 is the range of f. 4. The graphs of f and f -1 are reflections of each other in the line ________. 5. To show that two functions f and g are inverse functions, you must show that both f (g(x)) and g( f (x)) are equal to what? 6. Can (1, 4) and (2, 4) be two ordered pairs of a one-to-one function? 7. How many times can a horizontal line intersect the graph of a function that is one-to-one? 8. Give an example of a function that does not pass the Horizontal Line Test. Skills and Applications Finding an Inverse Function Informally In Exercises 9–16, find the inverse function of f informally. Verify that f ( f −1 (x)) = x and f −1 ( f (x)) = x. 9. f (x) = 6x 10. f (x) = 1 3 x 11. f (x) = 3x + 1 12. f (x) = x - 3 2 13. f (x) = x 3 + 1 14. f (x) = x 5 4 15. f (x) = x 2 - 4, x ≥ 0 16. f (x) = x 2 + 2, x ≥ 0 Verifying Inverse Functions In Exercises 17–20, verify that f and g are inverse functions algebraically. 17. f (x) = x - 9 4 , g(x) = 4x + 9 18. f (x) = - 3 2 x - 4, g(x) = - 2x + 8 3 19. f (x) = x 3 4 , g(x) = 3 √4x 20. f (x) = x 3 + 5, g(x) = 3 √x - 5 Sketching the Graph of an Inverse Function In Exercises 21 and 22, use the graph of the function to sketch the graph of its inverse function y = f −1 (x). 21. x 4 2 3 1 2 4 1 3 y 22. x 2 3 1 2 1 - 2 - 3 - 3 3 y Verifying Inverse Functions In Exercises 23–32, verify that f and g are inverse functions (a) algebraically and (b) graphically. 23. f (x) = x - 5, g(x) = x + 5 24. f (x) = 2x, g(x) = x 2 25. f (x) = 7x + 1, g(x) = x - 1 7 26. f (x) = 3 - 4x, g(x) = 3 - x 4 27.

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  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  A horizontal line reflected across the line y x = becomes a vertical line, and, therefore, the reflected graph of a one-to-one function passes the vertical line test. Recall that this is a test that determines whether a graph is the graph of a function. We conclude that the reflected graph of a one-to-one function is the graph of a function. Therefore, a one-to-one function f x ( ) is invertible, and the reflection of its graph about the line y x = is the graph of its inverse function. The notation we use for its inverse function is f x −1 ( ). x y x y × × × FIGURE 5.28. The horizontal line test. EXAMPLE 5.13.5. If f x a x ( )= , where a > 0 and ≠ a 1 , then f x x a −1 ( )= log . ± EXAMPLE 5.13.6. The function f x x ( ) = 2 3 + is a one-to-one function (its graph is the cubic graph shifted up two units); therefore, it has an inverse function f x −1 ( ). The graphs of y f x x = ( )= 2 3 + and y f x = ( ) 1 − are shown in figure 5.29. ± 154 • Foundations of Mathematics x y = f(x) y = f -1 (x) FIGURE 5.29. The inverse of a cubic function. Given a one-to-one function f x ( ) , there is the following method by which the explicit expres- sion for its inverse function f x −1 ( ) can be found: (i) Write the equation x f y = ( ) , that is, interchange x and y in the equation y f x = ( ) , using the explicit expression for f x ( ) ; (ii) then solve for y (if possible). The resulting expression is for f x −1 ( ) (if an explicit expression can be found). EXAMPLE 5.13.7. If f x x ( ) = 2 3 + , then, according to the method above, an expression for the inverse function f x −1 ( ) can be found by solving the equation x y = 2 3 + for y. The solution is y x = 2 3 − (check this). Therefore, f x x − − 1 3 ( ) = 2 , and the graph of y f x = ( ) 1 − is the graph of the cube root function shifted two units to the right (the reflected graph shown in figure 5.29). ± EXAMPLE 5.13.8. Real-valued functions that are strictly increasing or decreasing are one-to-one functions.

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