Math-3 Lesson 4-1 Inverse Functions

Definition A function is a set of ordered pairs with no two first elements alike. – f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) } But... what if we reverse the order of the pairs? – This is also a function... it is the inverse function – f -1 (x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }

Does This Have An Inverse? Given the function at the right – Can it have an inverse? – Why or Why Not? NO … when we reverse the ordered pairs, the result is Not a function – We would say the function is not one-to-one A function is one-to-one when different inputs always result in different outputs xY

Is the Inverse of a function a function? Horizontal line test: if a horizontal line passes through the graph of the relation at more than one location, then the inverse of that relation is NOT a function. Fails horizontal line test or The inverse relation is not a function.

One-to-One functions For every input there is exactly one output (the definition of a function) AND every output has exactly one input. More simply: a. b. c. Which function is “one-to-one” ? It passes both the horizontal and vertical line test.

One-to-One Functions Each input has exactly one output and each output has exactly one input. If the function passes the Horizontal line test, then its inverse is also a function.

Domain and Range The domain of f is the range of f -1 The range of f is the domain of f -1 Thus... we may be required to restrict the domain of f so that f -1 is a function

Restricting the domain of f(x) so that its inverse is a function. Cut the function into pieces using a vertical line so that it passes the horizontal line test. For x ≥ 2, you get the right ½ of the parabola which passes the horizontal line test. Rewrite the function so the it applies for only x-values x ≥ 2 g(x) and it’s inverse are inverses of each other.

Finding the Inverse: exchange the locations of ‘x’ and ‘y’ in the equation then solve for ‘y’.

Domain and Range Consider the function h(x) = x 2 – 9 Determine the inverse function Problem => f -1 (x) is not a function Rewrite the function so the it applies for only x-values x ≥ 0

Inverse Relations Inverse the two (4, 2) (x, y) = (2, 4) What is the pattern? A reflection across the line y = x. (1, -3) Inverse the two (x, y) = (-3, 1) Inverse Reflection Principle

Inverse Functions We’re not used to graphing ‘y’ as an input value, then finding the output value ‘x’ So…we can rewrite the equation as ‘y’ in terms of ‘x’ (it’s the same relation). Switch ‘x’ and ‘y’ Bottom line: inverse functions are reflections across the line y = x.

Is the Inverse Relation a Function? xy xy

Your Turn: Draw the following graph of: Is the inverse relation a function? On the same x-y plot draw

Inverse Function Defined If f(x) is a one-to-one function with Domain “D” and Range “R” then the inverse function of f(x), denoted Is a function whose Domain is “R” and whose Range Is “D” defined by: if and only if This is just saying the domain of a function is the range of its inverse function.

Natural Logarithm Function Exponential Function Domain = ? Range = ? Domain = ? Range = ?

Finding the Inverse function algebraicially Write the function in “y = “ format. Exhange ‘x’ and ‘y’ in the equation: Solve for ‘y’ to find the inverse function:

Finding Inverse Functions Algebraically Is this function “one-to-one”? 1.Rewrite the function so that ‘y’ is in terms of ‘x’. 3. Exchange ‘x’ and ‘y’ in the equation. Your turn: 5. Your turn: 5. Solve for ‘y’. x ≠ See if the inverse function exists by checking if f(x) is one-to-one. State any restrictions on the domain of f(x).

Solve for ‘y’. Finding inverse functions graphically can be easier Using the inverse reflection principle. Domain: x ≠ 1

Finding Inverse Functions (again) 1. Rewrite the function so that ‘y’ is in terms of ‘x’. 2. See if the inverse function exists by checking if f(x) is one-to-one. State any restrictions on the domain of f(x) to ensure that it is one-to-one. Passes horizontal line test. y ≥ 0 x ≥ -3 y ≥ 0 x ≥ -3

The domain of f(x) is the range of the inverse function. The range of f(x) is the domain of the inverse function. Finding Inverse Functions (again) 3. Switch the location of ‘x’ and ‘y’. 4. Solve for ‘y’. y ≥ -3x ≥ 0

Inverse functions:

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Verifying Inverse Functions Algebraically IF f(g(x)) = x (for every ‘x’ in the domain of g(x)) And IF g(f(x)) = x (for every ‘x’ in the domain of f(x) THEN: f(x) is a one-to-one function with inverse g(x)

Verifying Inverse Functions

Your Turn: Verify that the two functions are inverses of each other Verify that the two functions are inverses of each other.