5.6 Inverse Functions

Writing a formula for the input of a function Example 1      

What pattern do you see in the steps of each?   Step 1 Multiply by 2 Step 1 Subtract 3 Step 2 Add 3 Step 2 Divide by 2 What pattern do you see in the steps of each? Inverse Operations in Reverse Order These steps undo each other. Functions that undo each other are called inverse functions.

Inverse Functions   Original Function   Inverse Function     Add animations

Graphing Inverse Functions Original Function:   x -2 -1 1 2 y 3 5 7   Inverse Function: x -1 1 3 5 7 y -2 2 The graph of an inverse function is a reflection of the graph of the original function over the line y=x.

Finding the Inverse of a Linear Function Method 1 Use inverse operations in the reverse order   Multiply the input x by 3 and subtract 1 To find the inverse, apply inverse operations in the reverse order   Kinda conf. Add 1 to the input x then divide by 3

Finding the Inverse of a Linear Function (Cont.) Method 2             Add 1 to each side.   Divide each side by 3.

Example 2   Method #1 Method #2 Solve for y             Switch x and y        

Horizontal Line Test If the graph of a function y = f(x) is such that no horizontal line intersects the graph in more than one point then f is one to one and has an inverse function. Has an inverse function Has no inverse function

Inverses of Nonlinear Functions Notice: Inverses of linear functions were also functions. Inverse of a nonlinear function is not always a function. Same rule applies: The graph of an inverse is a reflection of the graph of the original function over the line y=x. Original Function:       x -2 -1 1 2 y 4 Inverse: (NOT A FUNCTION) x 4 1 y -2 -1 2

Example 3

Example 4                 Solve      

Example 5   Find and graph the inverse of the function.               Solve      

Example 6 Inverse Functions undo each other. This is why this definition makes sense: f(x) and g(x) are inverse functions if : 1)  g(f(x)) = x for all x in the domain of f 2)  f(g(x)) = x for all x in the domain of g Example 6

Example 7 Step 2: Evaluate the inverse when S = 100π Sometimes switching the variables in real-life problems create confusion, for example: Example 7 Step 2: Evaluate the inverse when S = 100π Step 1: Find the inverse To avoid a confusion, do not switch the variables, just solve for r The radius of the sphere is 5 feet.