Presentation on theme: "* Objectives: Students will be able to… * Determine whether a function has an inverse * Write an inverse function * Verify 2 functions are inverses of."— Presentation transcript:
* Objectives: Students will be able to… * Determine whether a function has an inverse * Write an inverse function * Verify 2 functions are inverses of each other
3 x f(x) y f -1 (x) x2x2 * The inverse of a given function will “undo” what the original function did. * For example, let’s take a look at the square function: f(x) = x 2
* The inverse of f is f -1 (read “f inverse”) * If both the original relation and the inverse relation are both functions, they are inverse functions! * The domain of the original relation is the range of the inverse.
Graphically, the x and y values of a function and its inverse are switched. If the function y = g(x) contains the points then its inverse, y = g -1 (x), contains the points Where is there a line of reflection?
The graph of a function and its inverse are mirror images about the line y = x y = f(x) y = f -1 (x) y = x
x y x y
Writing the equation of an inverse function: Example 1: y = 6x - 12 Example 1: y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:
* If f and g are inverse functions, their composition would simply give x back.
* Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses. * Because f(g(x))=x and g(f(x))=x, they are inverses. f(g(x))= -3(- 1 / 3 x+2)+6 = x-6+6 = x g(f(x))= - 1 / 3 (-3x+6)+2 = x-2+2 = x
Since both of these = x, if you start with x and apply the functions they “undo” each other and are inverses.
The graph of a function needs to pass the vertical line test in order to be a function.
inverse horizontal line test * Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test. passes inverse is a function * If the original function passes the horizontal line test, then its inverse will pass the vertical line test and therefore is a function. does not pass inverse is not a function (may need to restrict domain so that it has an inverse). * If the original function does not pass the horizontal line test, then its inverse is not a function (may need to restrict domain so that it has an inverse).
* All functions have only one y value for any given x value (this means they pass the vertical line test) * One-to-one functions also have only one x- value for any given y-value.
If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one- to-one function and will NOT have an inverse function. This is a one-to-one function This is NOT a one-to-one function
Graph does not pass the horizontal line test, therefore the inverse is not a function.
Graph y = x 2 for x > 0. Find the inverse function Graph y = x 2 for x > 0. Find the inverse function.
* How could I restrict the domain to make it have an inverse function? f -1 (x) is not a function. x > 0
Example 2: Given the function : y = 2x 2 -4, x > 0 find the inverse Step 1: Switch x and y: x = 2y 2 -4 Step 2: Solve for y: Only need the positive root for inverse!