Basics

2020 SAJC JC1 H2 Mathematics 1 Chapter 5: Functions Basics Fill in the blanks. Part 1: Definitions of Functions To test whether a relation is a function, we use the ________ __________________ __________________ It states that: If for each value a in ___________________, the vertical line x = a intersects the graph __________________, then the given graph is the graph of a function. Example of Relation that is a Function  f , , 0x x x x    f is a function because for each value fDa , i.e 0a  , the line x a _______________________________. Example of Relation that is not a Function  f , x x x   f is not a function because ___________ _________________________________ _________________________________ Part 2: Inverse Functions Existence of Inverse Function Only one-to-one function have an inverse. We use the _________ __________________ to check if the function is one-to-one, and thus, if the function has an inverse. It states that: a function f is one-to- one if __________ horizontal line f, Ry k k  intersects the graph of  fy x exactly once. Example of Function with Inverse   2f 1, , 1x x x x     Example of Function without Inverse   2f 1, , 2x x x x     x y x y IMPT!: Note the difference between proving and disproving. Disproving only requires a counter-example. y (1,2) x (-2,5) 1 y x y=3

2020 SAJC JC1 H2 Mathematics 2 Finding Rule of Inverse Function and Sketching Graph of its Inverse Consider the function:     2 f 1 2, , 1x x x x      . Verify that the inverse exists by sketching the graph and using the Horizontal Line Test. Sketch the graph of  fy x here. To find the inverse function, let y = f(x) and make x the subject.     2 2 1 2 1 2 1 2 1 2 y x x y x y x y             Since ____________________,   1 f 1 2, 2x x x     Sketch the graph of the inverse function. The graphs of  fy x and   1fy x  are reflections of each other in the line y x Use a few points to help guide you. Choose a different point on the graph  fy x and write down the corresponding point on the inverse graph.  fy x   1fy x  (-1,2) (-2, 3) y x y= x

2020 SAJC JC1 H2 Mathematics 3 Part 3: Composite Functions Finding Range of a Composite Function gf  f 2, 0 5x x x      2g , 4 4x x x    Alternative Method (not recommended if question did not ask for rule of gf or if gf is a complex function to sketch)       2 gf g 2 2 , 0 5 x x x x       3 -2 y x y 16 4 -4 x -2 3 9 y 4 2 5 9 x 5 Restrict the domain of g to only