Functions and Inverses Numerical Representation Consider the ordered pairs: (-3, -5) (-2, -3) (-1,-1) (0,1) (1,3) (2, 5) (3, 7) The inverse is found numerically by switching the x and y coordinates: (-5, -3) (-3, -2) (-1,-1) (1,0) (3,1) (5, 2) (7, 3)

Functions and Inverses Analytical Representation The general rule/relationship of the ordered pairs is: y = 2x + 1 y = 2x + 1 The inverse is found analytically by switching x and y in the equation and solving for y. Test out some ordered pairs to see that this works! This is just like finding the numerical inverse for all pairs

Functions and Inverses Graphical Representation The graph of y = 2x + 1 is seen in blue. The graph of y = 2x + 1 is seen in blue. The graph of is seen in red. The graph of is seen in red. A graph and its inverse are always symmetric about the line y =x

Example: Graphical to Numerical State the inverse of the function shown as a set of ordered pairs:

Graphical to Numerical We see that the original function has ordered pairs (-2,1)(-1,2)(2,-1) (3,1) and (4,4) Thus, the ordered pairs for the inverse are (1,-2) (2,-1)(-1,2)(1,3) and(4,4)

Example: Analytic (to Graphical) to Analytic Find the rule for the inverse of the following function: Find the rule for the inverse of the following function: y =

Example: Analytic (to Graphical) to Analytic Following the algorithm, students will likely say that the inverse is Which corresponds to However, a quick sketch of the graph will show us why this is not entirely correct….

The (red) inverse is not a reflection of the function over the line y = x There is an extra branch! We only want to use the part of the graph which is a reflection over the line y = x. This corresponds to the part of the red graph where the x coordinates are greater than or equal to zero.

The inverse of y= Is actually Without looking at the graph, most students will use both branches of the red graph, which is incorrect.

The previous conclusion could have also been reached numerically: Consider the ordered pair (-2, 4) which is a solution to the equation If (-2,4) makes the inverse equation true, then the point (4,-2) must make the original equation true, This is not true because square roots are defined to be positive values. Therefore, the inverse we came up with must not be entirely correct!

Numeric to Analytic State the inverse of the ordered pairs as an equation where y depends on x. Assume there are infinitely many ordered pairs which follow this pattern: (-2,-8) (-1,-1) (0,0) (1,1) (2,8) (3,27)(4,64)

Numeric to Analytic Students will need to determine the rule for the function which in this case is Now, analytically the students will find the inverse to be A quick sketch of the graph will show that no domain restrictions are necessary.

Numeric to Analytic State the inverse of the ordered pairs as an equation where y depends on x. Assume there are infinitely many ordered pairs which follow this pattern: (5,-5) (3,-3) (2,-2) (0,0) (1,1) (2,2) (5,5)

Numeric to Analytic Students often can not find a numerical representation for this relationship because it is not familiar. Encourage to represent the data graphically!

Numeric to Analytic Many students will recognize this graph as having the “absolute value shape”. So the relationship involve the absolute value function and can be represented as The question asked for the inverse of the rule, which is: y = |x| y = |x|

Choose the Best Representation Name a function with infinitely many ordered pair solutions which is its own inverse.

Stuck? This problem can easily be solved numerically because we are working with infinitely many ordered pairs.

Stuck? This problem can easily be solved analytically because there are too many functions to be able to easily guess and check.

Think Graphically! All we need is a function whose graph is symmetric about the line y = x!

Examples y = x y = x y = -x y = -x y = 1/x y = 1/x Use analytical, numerical, and graphical methods to verify that these equations are inverses of themselves!