 # 12.1 Inverse Functions For an inverse function to exist, the function must be one-to-one. One-to-one function – each x-value corresponds to only one y-value.

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12.1 Inverse Functions For an inverse function to exist, the function must be one-to-one. One-to-one function – each x-value corresponds to only one y-value and each y- value corresponds to only one x-value. Horizontal Line Test – A function is one-to- one if every horizontal line intersects the graph of the function at most once.

12.1 Inverse Functions f -1 (x) – the set of all ordered pairs of the form (y, x) where (x, y) belongs to the function f. Note: Since x maps to y and then y maps back to x it follows that:

12.1 Inverse Functions Method for finding the equation of the inverse of a one-to-one function: 1.Interchange x and y. 2.Solve for y. 3.Replace y with f -1 (x)

12.1 Inverse Functions Example: 1.Interchange x and y. 2.Solve for y. 3.Replace y with f -1 (x)

12.1 Inverse Functions Graphing inverse functions: The graph of an inverse function can be obtained by reflecting (getting the mirror image) of the original function’s graph over the line y = x

12.2 Exponential Functions Exponential Function: For a > 0 and a not equal to 1, and all real numbers x, Graph of f(x) = a x : 1.Graph goes through (0, 1) 2.If a > 1, graph rises from left to right. If 0 < a < 1, graph falls from left to right. 3.Graph approaches the x-axis. 4.Domain is: Range is:

12.2 Exponential Functions Graph of an Exponential Function (0, 1)

12.2 Exponential Functions Property for solving exponential equations: Solving exponential equations: 1.Express each side of the equation as a power of the same base 2.Simplify the exponents 3.Set the exponents equal 4.Solve the resulting equation

12.2 Exponential Functions Example: Solve: 9 x = 27

12.3 Logarithmic Functions Definition of logarithm: Note: log a x and a x are inverse functions Since b 1 = b and b 0 = 1, it follows that: log b (b) = 1 and log b (1) = 0

12.3 Logarithmic Functions Logarithmic Function: For a > 0 and a not equal to 1, and all real numbers x, Graph of f(x) = log a x : 1.Graph goes through (0, 1) 2.If a > 1, graph rises from left to right. If 0 < a < 1, graph falls from left to right. 3.Graph approaches the y-axis. 4.Domain is: Range is:

12.3 Logarithmic Functions Graph of an Exponential Function Try to imagine the inverse function

12.3 Logarithmic Functions Inverse - Logarithmic Function

12.3 Logarithmic Functions Example: Solve x = log 125 5 In exponential form: In powers of 5: Setting the powers equal:

12.4 Properties of Logarithms If x, y, and b are positive real numbers where Product Rule: Quotient Rule: Power Rule: Special Properties:

12.4 Properties of Logarithms Examples: Product Rule: Quotient Rule: Power Rule: Special Properties:

13.1 Additional Graphs of Functions Absolute Value Function Graph of What is the domain and the range?

13.1 Additional Graphs of Functions Graph of a Square Root Function Graph of (0, 0)

13.1 Additional Graphs of Functions Graph of a Greatest Integer Function Graph of Greatest integer that is less than or equal to x

13.1 Additional Graphs of Functions Shifting of Graphs Vertical Shifts: The graph is shifted upward by k units Horizontal shifts: The graph is shifted h units to the right If a < 0, the graph is inverted (flipped) If a > 1, the graph is stretched (narrower) If 0 < a < 1, the graph is flattened (wider)

13.1 Additional Graphs of Functions Example: Graph Greatest integer function shifted up by 4

13.1 Additional Graphs of Functions Composite Functions Composite function: function of a function f(g(x)) = (f  g)(x) Example: if f(x) = 2x – 1 and g(x) = x 2 then f(g(x)) = f(x 2 ) = 2x 2 – 1 What is g(f(2))? Does f(g(x)) = g(f(x))?

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