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Published byGalilea Walsingham Modified over 2 years ago

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* 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

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3 x f(x) 3 3 3 3 3 9 9 9 9 9 9 9 y f -1 (x) 9 9 9 9 9 99 3 3 3 3 3 3 3 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

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* 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.

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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?

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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

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x-35710 y-5-3047 x y

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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:

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1.2.

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* If f and g are inverse functions, their composition would simply give x back.

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* 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

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Since both of these = x, if you start with x and apply the functions they “undo” each other and are inverses.

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The graph of a function needs to pass the vertical line test in order to be a function.

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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).

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* 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.

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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

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Graph does not pass the horizontal line test, therefore the inverse is not a function.

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Graph y = x 2 for x > 0. Find the inverse function Graph y = x 2 for x > 0. Find the inverse function.

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* How could I restrict the domain to make it have an inverse function? f -1 (x) is not a function. x > 0

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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!

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Inverse is a function! y=2x 3 x=2y 3

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