Relations are a pairing of input and output values. You could say a list of points.

Functions pass the vertical line test. They have only one y-value for each x-value.

B A C How do you know?

Starburst Hershey’s Skittles Snickers Reese’s Input Output Notice there is more than one output value for the input values of 1 and of 2. ( Vertical Line Test)

Starburst Hershey’s Skittle s Snicker s Reese’s Input Output Starburst Hershey’ s Skittles Snickers Reese’s Input Output How do you know? AB

The inverse has all of the x’s and y’s reversed.

f(x) = {(-2,0),(-1,3),(0,6),(1,0),(2,12)} Find f -1 (x) f -1 (x) = {(0,-2), (3,-1),(6,0),(0,1),(12,2)} Is the inverse a function? The -1 means the inverse of f(x) NO

1) y = 2x + 3 What should we do to find the inverse? Exchange the x & y values x = 2y + 3 Isolate the “y”. 2y = x – 3 Is the inverse a function? YES If the original problem is “linear” then the inverse is linear. Both are functions.

2) y = 6 Exchange the x & y values x = 6 Is the inverse a function? NO, because it is a vertical line. Write your answer like this: The inverse is x = 6 When the original problem is a horizontal line (yes a function) the inverse is a vertical line (not a function).

4) y = x Find the inverse NEW shape: We are starting with a quadratic function. Original problem “is” a function. Inverse “is NOT” a function.

 Plus or minus a square root function.

4) y = x Was the inverse a function? Would the original problem have passed a horizontal line test? Then the inverse will not pass a vertical line test. NOT a function. NO

 y = 2x – 4  y = 9  y = x Would the given problem pass a horizontal line test? Will the inverse be a function?

The inverse on a graph is reflected across the line y = x Or you could say a 45 0 angle with the x-axis. HOW did I do that so quickly? List critical points and switch the x and y.

1) Pick “critical points” on the given graph. 2) Switch the x and y values. 3) Sketch the new graph, in order. 4) Fold your paper to be sure you are correct.

Know the difference between relations and functions. Be able to find the inverse of a set of points (ordered pairs). Be able to take a linear or quadratic function and algebraically find the inverse. Be able to draw the inverse of a graph.

Does every function have an inverse? Is the inverse of every function also a function? What shape is the inverse of a quadratic function? What shape is the inverse of a linear function? What does the vertical line test tell you? What does the horizontal line test tell you?

WS