100921 Notes Inverses of functions

Making a list of points for each point a b included

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Making a list of points: For each point ( a , b ) included in f ( x ), put ( b , a ) in the inverse. Making a graph: For each point ( a , b ) on the graph of f ( x ), include the point ( b , a ) on the inverse graph. Making a graph using reflection: Take the graph of f ( x ) and draw its “mirror image” reflected across the diagonal line y = x . Using equations: Start with the f ( x ) formula written in the x and y notation. Form the inverse by changing all x ’s to y ’s and all y ’s to x ’s. Then, if possible, rewrite the equation so that it is solved for y . If you make an inverse, will it be a function too? To get an inverse that is also a function, the function you start with has to be a one-to-one function . This means: In terms of inputs and outputs: For every input there is only one output, and for every output there is only one input . In terms of x and y values: For every x there is only one y , and for every y there is only one x . In terms of tables: There are no repeated values in the input column and no repeated values in the output column . In terms of graphs: The graph passes both the “vertical line test” and the “horizontal line test.” The underlined part of the requirement is the part that goes beyond what’s required just to be a function.
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Name Honors Algebra 2 (J. Sera) Date _________________________ Class notes Inverse function notation If f ( x ) is a one-to-one function, the name f –1 ( x ) is used for its inverse function. For example, for f ( x ) = x + 5, the inverse is f –1 ( x ) = x – 5. Important: The –1 is just the symbol for inverse ; it does not represent an exponent or a power; also it has nothing to do with the calculator key.
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