Finding the Inverse

 If f(a) = b, then a function g(x) is an inverse of f if g(b) = a.  The inverse of f(x) is typically noted f -1 (x), which is read “f inverse of x” so equivalently, if f(a) = b, then f -1 (b) = a  Our inputs and outputs switch places  Important: The raised -1 used in the notation for inverse functions is simply a notation and does not designate an exponent or power of -1.

 If f(2) = 4, what do we know about the inverse?  If f(2) = 4, then f -1 (4) = 2.

 If h -1 (6) = 2, what do we know about the original function, h(x)?

 Using the table below, find and interpret the following:  A. f(60)  B. f -1 (60) t(minutes) F(t) (miles)

1 st example, begin with your function f(x) = 3x – 7 replace f(x) with y y = 3x - 7 Interchange x and y to find the inverse x = 3y – 7 now solve for y x + 7 = 3y = y f -1 (x) = replace y with f -1 (x)

2 nd example g(x) = 2x replace g(x) with y y = 2x Interchange x and y to find the inverse x = 2y now solve for y x - 1 = 2y 3 = y 3 = y g -1 (x) = replace y with g -1 (x)

 Not all functions will have an inverse function.  A function must be a one to one function to have an inverse.  To verify if two functions are inverses of on another, you can check the composition of functions with the inverse. If both solutions equal x, they are inverses.

Consider f(x) = What is the domain? x + 4 > 0 x > -4 or the interval [-4, ∞) What is the range? y > 0 or the interval [0, ∞)

Now find the inverse: f(x) =D: [-4, ∞) R: [0, ∞) y = Interchange x and y x = x 2 = y + 4 x 2 – 4 = y f -1 (x) = x 2 – 4D: [0, ∞) R: [-4, ∞)

Finally, let us consider the graphs: f(x) = D: [-4, ∞) R: [0, ∞) blue graph f -1 (x) = x 2 – 4 D: [0, ∞) R: [-4, ∞) red graph

2 nd example Consider g(x) = 5 - x 2 D: [0, ∞) What is the range? Make a very quick sketch of the graph R: (-∞, 5]

Now find the inverse: g(x) = 5 - x 2 D: [0, ∞) R: (-∞, 5] y = 5 - x 2 Interchange x and y x = 5 - y 2 x – 5 = -y 2 5 – x = y 2 = y but do we want the + or – square root? g -1 (x) = D: (-∞, 5] R: [0, ∞)

And, now the graphs: g(x) = 5 - x 2 D: [0, ∞) R: (-∞, 5] blue graph g -1 (x) = D: (-∞, 5] R: [0, ∞) red graph

A function is one-to-one if each x and y-value is unique Algebraically it means if f(a)=f(b), then a=b. On a graph it means the graph passes the vertical and the horizontal line tests. If a function is one-to-one it has an inverse function.