THE INVERSE GRAPH. Finding the equation of the inverse. What is a function? LESSON

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1 LESSON THE INVERSE GRAPH The reflection of a graph in the line = will be the graph of its inverse. f() f () The line = is drawn as the dotted line. Imagine folding the page along the dotted line, the two circles above would coincide. The following table shows corresponding points on the two circles. Circle centre (;) Circle centre (;) (;) (;) (;) (;) (;) (;) (-;) (;-) Notice that ever time, the and values at the point are simpl interchanged. Finding the equation of the inverse Step : Swap the and Step : Make the subject of the formula, if required. In the above eample, the equation of the circle centre on the -ais is given b + ( ) =. To find the equation of the inverse graph of this circle: swap and : + ( ) = It is not necessar to make the subject for the circle equation, but for all functions we do. What is a function? A function is a relation(graph) where each is linked to onl one value. The circles above are not functions. Look at the circle with centre on the -ais. When =, = or =. Thus with =, there are two values of linked to it. Page 7

2 A quick and eas wa to check whether a graph is a function or not, is to do the vertical line test. Vertical Line Test:. Draw a vertical (imaginar) line through the graph.. If all vertical lines cut the graph onl once, the graph is a function. In the circles above, there eist man vertical imaginar lines that cut the circle in points. Therefore the circle is not a function. The graphs labeled A,B and C are all functions. (since vertical lines onl cut the graph in one point). B A 5 C The graphs labeled D,E, F and G below are not functions (since an vertical line can cut the graphs at more than one point). F G D E Notice how the vertical lines cut the graphs in more than one point. What is an INVERSE FUNCTION? An inverse function is an inverse that is also a function. Eample _ The graph of = and its inverse = ±. Eample 9 Page Lesson Algebra Page 7

3 = 6 = To find the equation of the inverse of =, Swap the and : = _ Make the subject of the formula: = ± The inverse of = is not a function. Eample Eample The graph of = and its inverse function = log : = To find the equation of the inverse of =, swap the and : = make the subject of the formula: = log The graph of = has an inverse function. Page 7

4 Eample The graph of = + is sketched, together with its inverse. = + Eample = _ _ 5 To find the equation of the inverse: Swap the and the : = + Make the subject: = \ = _ _ An straight line graph will have an inverse that is also a function, i.e. = m + c will have an inverse function. Activit Activit. For each of the graphs sketched below, state whether the inverse graph will be a function or not. Give reasons.. 9 Page Lesson Algebra Page 7

5 . 6. For each of the following graphs, draw the inverse graph on the same set of aes C A B 6 5 Page 7

6 . 6. Given: f() =. Find the point(s) of intersection of f and f, the inverse of f.. On the same set of aes sketch the graphs of f and f. Determine the equation of f in the form f :. Given: The graph of g() = a, where a >, passes through the point (; _ ).. Determine the value of a. 9 Page Lesson Algebra Page 75

7 . Determine the equation of g, the inverse of g, in the form =. Sketch the graphs of g and g on the same set of aes. 5. Given: f() = _ ; ³ 5. Find the point(s) of intersection of f and f, the inverse of f. 5. Sketch the graphs of f and f on the same set of aes 5. Determine the equation of f in the form f () = Page 76

8 Solutions. Inverse is not a function. Consider the points (; ) and (; ) on the graph. The corresponding points on the inverse will be (; ) and (; ). Thus for = there is more than one value.. Inverse is a function. The graph and the inverse have one -value for each -value. This is also known as a one to one function. Therefore the inverse will be one to one. If a graph is one to one then its inverse will also be one to one.. Yes. The graph is one to one. Therefore the inverse will be one to one.. = 5. C = A B 5 B C A 9 Page Lesson Algebra Page 77

9 .. =. \ = = () point of intersection (;) = f = If f is a one-to-one function, then the point of intersection between f and f is alwas along the line = the - and -co-ordinate values are alwas the same. f. For inverse, = + \ = \ = _ _ \ f : _ _. = a \ _ = a \ a = _. For the inverse, = ( _ ) = lo g _ (because f is a one-to-one function) Page 78

10 g = g _ = (because f is a one-to-one function) \ = 5. \ + = \ ( + ) = \ = or = (N.A) Therefore, the point(s) of intersection are (; ) f 6 5 = 5. For the inverse: (swap and ) = _ ; \ = ; f \ = ; _ = ± ; _ \ f () = ± ; (Note:, since we cannot square root a negative.) Note: The restriction must be included because f - is the inverse of a restriction on f. 9 Page Lesson Algebra Page 79