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Published byLjiljana Π‘ΡΠ°Π½ΠΎΡΠ΅Π²ΠΈΡ Modified over 5 years ago
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Warm Up Chain Reaction Choose one team member to start problem #1.
After the first person evaluates π π₯ , pass the paper to the right. The next team member evaluates π π π₯ . Keep passing the paper to the right until you have a solution. Have a different team member start problem #2. π π₯ =2π₯ g π₯ =π₯+6 β π₯ = βπ₯+10 2 π π₯ = π₯ 2
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π₯ 2 β4π₯+4 π(β(π(π(π₯)))) π₯ 2 β4π₯+1 π(β(π(π π₯+1 ))) π(β(π(π 2β π₯ 2 )))
Warm Up Chain Reaction Working through the functions in order, find: π(β(π(π(π₯)))) π₯ 2 β4π₯+4 π₯ 2 β4π₯+1 π(β(π(π π₯+1 ))) π(β(π(π 2β π₯ 2 ))) π₯ 4 π(β(π(π 2π₯ ))) 12 π₯ 2 β4π₯ 3 +1
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Quiz Results 2nd Period Average: 94.8% Median: 29 = 96.7% 3rd Period Average: 94.0% 4th Period Average: 92.0% Median: 28.5 = 95%
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Section 4-5 Inverse Functions
Objective: To find the inverse of a function, if the inverse exists. Inverse Definition Finding the Inverse Algebraically Graphing the Inverse Horizontal Line Test: One to one Function Domain & Range
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Functions Imagine functions are like the dye you use to color eggs. The white egg (x) is put in the function blue dye, B(x), and the result is a blue egg (y).
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The Inverse Function βundoesβ what the function does.
The Inverse Function of the Blue dye is bleach. The bleach will βundyeβ the blue egg and make it white.
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For example, letβs take a look at the square function: f(x) = x2
In the same way, the inverse of a given function will βundoβ what the original function did. For example, letβs take a look at the square function: f(x) = x2 x f(x) y π βπ (π) 9 3 3 9 9 3 3 9 9 3 3 9 9 3 3 x2 9 9 3 3 9 9 9 3 3 3 9 9
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For example, letβs take a look at the square function: f(x) = x2
In the same way, the inverse of a given function will βundoβ what the original function did. For example, letβs take a look at the square function: f(x) = x2 x y π βπ (π) f(x) 5 25 5 5 5 25 25 5 5 25 25 5 5 x2 25 5 5 25 5 25 25 5 25 5 5 5
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Inverse Function Definition
The inverse of a function f is written π β1 and is read βf inverseβ π β1 (π₯) is read, βf inverse of xβ Inverse Function Definition Two functions f and g are called inverse functions if the following two statements are true: 1. π(π π₯ )= π₯ for all x in the domain of f. 2. π(π π₯ )=π₯ for all x in the domain of g.
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π(π₯)=2π₯ +1 π π π₯ =π π π₯ =π₯ Example
Consider the functions f and g listed below. Show that f and g are inverses of each other. π(π₯)=2π₯ +1 Show that: π π π₯ =π π π₯ =π₯
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Find the inverse of a function algebraically:
Given the function: f(x) = 3x Find the inverse. *Note: You can replace f(x) with y. x = 3y2 + 2 Step 1: Switch x and y Step 2: Solve for y π βπ π = πβπ π
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Graphically, the x and y values of a point are switched.
If the function y = g(x) contains the points x 1 2 3 4 y 8 16 then its inverse, y = g-1(x), contains the points x 1 2 4 8 16 y 3 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
π = π(π) π = π π = π βπ (π) y = x
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Vertical and Horizontal Line Test
Does the graph pass the vertical line test? Does the graph pass the horizontal line test? What does passing/not passing the horizontal line test mean? π π = π β ππ
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The Vertical Line Test If the graph of π¦ = π(π₯) is such that no vertical line intersects the graph in more than one point, then f is a function.
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No! Yes! No! Yes!
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On the same axes, sketch the graph of
and its inverse. Notice Solution: x
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On the same axes, sketch the graph of
and its inverse. Notice Solution: Using the translation of what is the equation of the inverse function?
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Domain and Range The previous example used The Domain of is . Since is found by swapping x and y, the values of the Domain of give the values of the range of Domain Range
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Domain and Range The previous example used The domain of is Since is found by swapping x and y, the values of the domain of give the values of the range of Similarly, the values of the range of give the values of the domain of
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GRAPHING SUMMARY The graph of is the reflection of in the line y = x. At every point, the x and y coordinates of become the y and x coordinates of The values of the domain and range of swap to become the values of the range and domain of
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π π =β π+π π βπ State the domain and range of π(π₯).
π π =β π+π π βπ State the domain and range of π(π₯). Is π π₯ one-to-one? State your reason and the implication of a βyesβ or β noβ answer. Find the equation for π(π₯) β1 . Restrict the domain if necessary. Make sure to state the restricted domain. State the domain and range of π(π₯) β1 Graph π π₯ and π(π₯) β1 on the same grid. Show π π βπ π = π βπ (π π )=π
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Homework Page 149 #1-27 odds, 30
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