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5.3 Inverse Function

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**After this lesson, you should be able to:**

Verify that one function is the inverse function of another function. Determine whether a function has an inverse function. Find the derivative of an inverse function.

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**Definition of a Real-Valued Function of a Real Variable**

Review Definition of a Real-Valued Function of a Real Variable Function is a mapping!

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**Review Relation – a set of ordered pairs.**

Function – a set of ordered pairs in which no two ordered pairs have the same x-value and different y-values. A function is a rule or correspondence which associates to each number x in a set X a unique number f(x) in a set Y Both relation and function are a mapping. A function is a relation but not vise versa. Function Relation

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**Example 1 Tell the following relations are function or not:**

1. {(1, 0), (-2, 3), (3, -1), (1, -2), (3, 0)} Not a function 2. {(2, 0), (-5, 0), (3, 0)} Function 3. {(0, -1), (-2, 3), (4, -1), (1, -2), (-6, -2)} Function 4. {(1, 0), (3, 2)} Function 5. {(1, 0), (0, 3), (3, -1), (1, 1), (6, -2), (9, 0)} Not a function What is the characteristic of the non-functions? At least two points on the same vertical line!

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**Vertical Line Test (a graphical test for a function) See P. 22**

A graph is a function graph if and only if every vertical line intersects the graph at most ONCE.

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**Definition of Inverse Function and Figure 5.10**

X x2 x1 f f –1 y2 y1 f (X)

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**What kind of function has inverse function? Revisit Example 1**

Example 1 Tell the following relations are function or not: 2. {(2, 0), (-5, 0), (3, 0)} Function No inverse 3. {(0, -1), (-2, 3), (4, -1), (1, -2), (-6, -2)} Function No inverse 4. {(1, 0), (3, 2)} Function Has inverse X x2 x1 f f –1 y2 y1 f (X)

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**What kind of function has inverse function?**

Answer: A function has inverse must have the following properties: For any two different x-values, their images must be different: If x1 ≠ x2 , then f(x1) ≠ f(x2). For any y-value in f(X), there exists an original image in X: For any y ∈ f(X), there exists x ∈ X such that y = f(x). This kind of function is called one-to-one function. Only one-to-one function has inverse function. How do we know a function is a one-to-one function by graph?

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**Horizontal Line Test (a graphical test for a one-to-one function) See P. 343**

A function has an inverse if and only if every horizontal line intersects the graph of a function at most ONCE.

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Example 2 The function y = x2 – 4x + 7 is not one-to-one on the real numbers because the line y = 7 intersects the graph at both (0, 7) and (4, 7). x y 2 (0, 7) y = 7 (4, 7)

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**Example: Horizontal Line Test**

Example 3 Apply the horizontal line test to the graphs below to determine if the functions are one-to-one. a) y = x3 b) y = x3 + 3x2 – x – 1 x y -4 4 8 x y -4 4 8 one-to-one not one-to-one Example: Horizontal Line Test

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**Theorem 5.7 The Existence of an Inverse Function**

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**Theorem 5.6 Reflective Property of Inverse Functions and Figure 5.12**

The domain of the inverse relation is the range of the original function. The range of the inverse relation is the domain of the original function.

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**What is the relationship between the graph of the function and the graph of its inverse function?**

Their graphs are symmetry to the line y = x

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**Example: Determine Inverse Function**

Example 4 From the graph of the function y = f (x), determine if the inverse function exists and, if it does, sketch the graph of inverse. The graph of f passes the horizontal line test. y y = f -1(x) y = x y = f(x) The inverse function exists. x Reflect the graph of f in the line y = x to produce the graph of f -1. Example: Determine Inverse Function

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**Guidelines for Finding an Inverse Function**

Note We should add 1a. into to the guideline 1a. Find the domain and range of the f

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**Verify two functions are inverse each other algebraically**

Example 5 Find the inverse of the following function Solution Graphical test show this function has inverse Domain and range of this function are R, R. Domain and range of the inverse function are R, R.

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Continued… Verify the inverse function algebraically.

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**Find the inverse algebraically**

Example 6 Find the inverse of the following function Solution Graphical test show this function has inverse Domain and range of this function are R, R. Note that and So,

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Continued… Domain and range of the inverse function are R, R. Verify the inverse function algebraically.

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Continued…

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**Theorem 5.8 Continuity and Differentiability of Inverse Functions**

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**We now discuss the relationship between the derivative of the function and its inverse.**

Suppose that a function and its non-zero derivative are And then its inverse is Taking the derivative of the inverse with respect of variable y, we have or

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**Since the variable y in this expression is only the dummy variable, so we change y to x.**

The above is not a formal proof.

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**Application of Derivative of Inverse Function**

Example 7 Find the derivative of function Solution It is kind of hard to find the derivative of g directly. Let’s consider another function It is easy to know that g and f are inverse each other. And we know the derivative of f in the previous section. So By using the concept in the above example, we can find the derivatives for many functions.

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**Application of Derivative of Inverse Function**

Example 8 Let a. What is the value of when b. What is the value of when Solution Notice that f is one-to-one function and therefore has its inverse f -1. a So b. By the Theorem 5.9, we know

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**Application of Derivative of Inverse Function**

Example 9 Let (for x ≥ 0) and let Show that the slopes of graphs of f and f -1 are reciprocal at each of the following points: (a, a2) and (a2, a) (a>0) Solution The derivatives of f and f -1 are: and At point (a, a2), the slope of graph of f is At point (a2, a), the slope of graph of f -1 is

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Homework Pg odd, 33, 35, odd

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