5.1 Inverse Functions
5.1 Inverse Functions
One-to-One Functions * One-to-one function is where each x-value corresponds to one y-value, and each y-value corresponds to only one x-value * * Horizontal line test – a function is one-to-one if every horizontal line intersects the graph of the function at most once * Examples: Determine whether the following functions are one-to-one
* In general, a function that is either increasing or decreasing on its entire domain, such as must be one-to-one * Tests to Determine Whether a Function is One-to-One * Show that f(a) = f(b) implies a = b; then the function is one-to-one * Every y-value corresponds to no more than one x-value; to show that the function is not one-to-one, find at least two x-values that produce the same y-value * Use the horizontal line test * If the function either increases or decreases on its entire domain, then it is one-to-one
Inverse Functions * Let be a one-to-one function. Then g is the inverse function of if: * * Examples: Determine whether f and g are inverses * * * * Inverse of a function is denoted as * By definition of inverse function, the domain of f is the range of and the range of f is the domain of. * To find the inverse of a one-to-one function, interchange the x- and y-values of each of its ordered pairs. * Functions that are not one-to-one do not have an inverse
* Examples: If the functions are one-to-one, find their inverses. *
* Steps for Finding the Equation of the Inverse of a Function * Interchange x and y * Solve for y * Replace y with * Examples: For each function that is one-to-one, a) write an equation for the inverse function in the form , b) graph the function & its inverse, c) give the domain and range of the
One-to-One Functions * One-to-one function is where each x-value corresponds to one y-value, and each y-value corresponds to only one x-value * * Horizontal line test – a function is one-to-one if every horizontal line intersects the graph of the function at most once * Examples: Determine whether the following functions are one-to-one
* In general, a function that is either increasing or decreasing on its entire domain, such as must be one-to-one * Tests to Determine Whether a Function is One-to-One * Show that f(a) = f(b) implies a = b; then the function is one-to-one * Every y-value corresponds to no more than one x-value; to show that the function is not one-to-one, find at least two x-values that produce the same y-value * Use the horizontal line test * If the function either increases or decreases on its entire domain, then it is one-to-one
Inverse Functions * Let be a one-to-one function. Then g is the inverse function of if: * * Examples: Determine whether f and g are inverses * * * * Inverse of a function is denoted as * By definition of inverse function, the domain of f is the range of and the range of f is the domain of. * To find the inverse of a one-to-one function, interchange the x- and y-values of each of its ordered pairs. * Functions that are not one-to-one do not have an inverse
* Examples: If the functions are one-to-one, find their inverses. *
* Steps for Finding the Equation of the Inverse of a Function * Interchange x and y * Solve for y * Replace y with * Examples: For each function that is one-to-one, a) write an equation for the inverse function in the form , b) graph the function & its inverse, c) give the domain and range of the