Inverse Functions

Review Let’s draw a flow chart of f(g(x)) Review Let’s draw a flow chart of f(g(x)). Now let f(x) = x2 + 1 and g(x) = √x. Compute and graph f(g(x)). Did you get f(g(x)) = x + 1? What is the domain? What impact does this have on the graph?

Graph f(x) = 3x. x y 1 2 Now switch x and y.

x y 0 0 3 1 6 2 Graph this function. Call this function g(x). What is the equation for g(x)? G(x) = x 3

Some things to note about f(x) and g(x): x and y are interchanged they “undo” each other they are reflections in the line y = x The domain of f is the range g and the range of f is the domain of g. f and g are called inverse functions

Can you think of other functions that “undo” each other?

Two functions f and g are inverse functions if f(g(x)) = g(f(x)) = x for all values of x in the domains Show that this is true for f and g.

Show that f(x) = ½x + 5 and g(x) = 2x – 10 are inverse functions. Can you understand how these two functions “undo” each other? How could you find an inverse function if it were not given?

Finding inverse functions Switch x and y Solve for y Rewrite in function notation Find the inverse of f(x) = ½x + 5 using this method. Find the inverse of f(x) = √(x+ 1) Find the inverse of f(x) = 3x + 1 x - 4

The inverse of a function f(x) is given by the notation f -1 (x) The inverse of a function f(x) is given by the notation f -1 (x). (The negative one is not an exponent. It does not mean 1/f(x)!)

Now try these… If f and g are inverses, and f(3) = -1, what is the g(-1)? If f(x) = 2x – 5, find f(f -1(2)). Draw the inverse of f(x) = x3+ 1.

Does f(x) = x2 have an inverse function? (Think about the graph.) To have an inverse a function must be one-to-one, which means that no two elements in the domain can correspond to the same element in the range. Sometimes we use the horizontal line test to check for this.

Let f(x) = x2+ 1. This is not one-to-one Let f(x) = x2+ 1. This is not one-to-one. If we stipulate that x ≥ 0, find f -1(x). f -1(x) = √(x-1). Look at the graphs to make sure they are reflections in y = x. What are the domain and range for each function?