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1.6 Inverse Functions Students will find inverse functions informally and verify that two functions are inverse functions of each other. Students will use graphs of functions to decide whether functions have inverse functions. Students will determine if functions are one-to-one. Students will find inverse functions algebraically.

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**Logical inverse of a statement:**

Statement: Put your coat on and go outside. An inverse function will reverse the order and action of the statement. It will also be the logical event that would undo the original statement. Inverse Statement:

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**Logical inverse of a function:**

Equation: An inverse of a function will undo every action on the variable. Consider what is being done to the variable along with the order. To find the inverse, reverse the order and the action taken on the variable. Inverse Equation: To algebraically solve for the inverse of a function, switch the domain and the range (switch the x and y) and solve for y. Consider what is being done when applying this method compared to the logical method. *Switching the domain and the range of a function will give the inverse.

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**Expressing Inverses: Verifying Inverses:**

If is a function and is the inverse of the function then Verifying Inverses: If f(x) and g(x) are inverse functions then f(g(x)) = x and g(f(x)) = x. Check this for g and f.

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Example 3: Show that the functions are inverse functions of each other.

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**Example 5: *Inverse Functions will reflect over the line y = x.**

Verify that the functions f and g from example 3 are inverse functions of each other graphically. *Inverse Functions will reflect over the line y = x.

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**Existence of an Inverse Function**

In order for an equation to represent a function, it must pass the vertical line test. In order for its inverse to be a function, what test would you think it must pass in order to determine if it is a function? Remember in order to find the inverse you must switch the domain and the range of a function. Remember the soda machine scenario. When mapping a function, it was OK to have 2 to 1 but not OK to have 1 to 2. When finding the inverse of a function the arrows switch. So what kind of function must we have in order to have an inverse?

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Example 6: Is the function one-to-one?

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Example 9 Find the inverse function of and use a graphing utility to graph and in the same viewing window.

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Do Now: Find f(g(x)) and g(f(x)). f(x) = x + 4, g(x) = x 2 + 2 f(x) = x + 4, g(x) = x 2 + 2.

Do Now: Find f(g(x)) and g(f(x)). f(x) = x + 4, g(x) = x 2 + 2 f(x) = x + 4, g(x) = x 2 + 2.

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