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Section5.1
Inverse Functions
Section5.1 Inverse Functions One-To-OneFunctions Definition A - - PowerPoint PPT Presentation
Section5.1 Inverse Functions One-To-OneFunctions Definition A - - PowerPoint PPT Presentation
Section5.1 Inverse Functions One-To-OneFunctions Definition A one-to-one (or 1-1 ) function has the property that no two x -values (inputs) have the same y -value (output). Definition A one-to-one (or 1-1 ) function has the property that no two
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Definition
A one-to-one (or 1-1 ) function has the property that no two x-values (inputs) have the same y-value (output).
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Definition
A one-to-one (or 1-1 ) function has the property that no two x-values (inputs) have the same y-value (output). Graphically, one-to-one functions pass the Horizontal Line Test : every horizontal line can intersect the graph at most once. One-to-One Not One-to-One
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Definition
A one-to-one (or 1-1 ) function has the property that no two x-values (inputs) have the same y-value (output). Graphically, one-to-one functions pass the Horizontal Line Test : every horizontal line can intersect the graph at most once. One-to-One Not One-to-One
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Examples
Which graphs correspond to one-to-one functions? 1.
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Examples
Which graphs correspond to one-to-one functions? 1. One-to-One
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Examples
Which graphs correspond to one-to-one functions? 1. One-to-One 2.
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Examples
Which graphs correspond to one-to-one functions? 1. One-to-One 2. Not One-to-One
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Examples
Which graphs correspond to one-to-one functions? 1. One-to-One 2. Not One-to-One 3.
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Examples
Which graphs correspond to one-to-one functions? 1. One-to-One 2. Not One-to-One 3. One-to-One
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InverseFunctions
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Definition
The inverse function of a function f is the function that switches all the inputs and outputs. In other words, if (a, b) is a point on the graph of f , then (b, a) is a point of the graph of f −1.
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Definition
The inverse function of a function f is the function that switches all the inputs and outputs. In other words, if (a, b) is a point on the graph of f , then (b, a) is a point of the graph of f −1. Graphically, f and f −1 are reflections of each other across the diagonal line y = x.
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Definition
The inverse function of a function f is the function that switches all the inputs and outputs. In other words, if (a, b) is a point on the graph of f , then (b, a) is a point of the graph of f −1. Graphically, f and f −1 are reflections of each other across the diagonal line y = x. For example, consider f (x) = 2x − 3:
−6 −4 −2 2 4 6 −10 −5 5
x f (x)
- 2
- 7
- 3
2 1
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Definition
The inverse function of a function f is the function that switches all the inputs and outputs. In other words, if (a, b) is a point on the graph of f , then (b, a) is a point of the graph of f −1. Graphically, f and f −1 are reflections of each other across the diagonal line y = x. For example, consider f (x) = 2x − 3:
−6 −4 −2 2 4 6 −10 −5 5
x f (x)
- 2
- 7
- 3
2 1
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Definition
The inverse function of a function f is the function that switches all the inputs and outputs. In other words, if (a, b) is a point on the graph of f , then (b, a) is a point of the graph of f −1. Graphically, f and f −1 are reflections of each other across the diagonal line y = x. For example, consider f (x) = 2x − 3:
−6 −4 −2 2 4 6 −10 −5 5
x f (x)
- 2
- 7
- 3
2 1
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Definition
The inverse function of a function f is the function that switches all the inputs and outputs. In other words, if (a, b) is a point on the graph of f , then (b, a) is a point of the graph of f −1. Graphically, f and f −1 are reflections of each other across the diagonal line y = x. For example, consider f (x) = 2x − 3:
−6 −4 −2 2 4 6 −10 −5 5
x f (x)
- 2
- 7
- 3
2 1 x f −1(x)
- 7
- 2
- 3
1 2
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Functions without Inverses
Not every function has an inverse function, however: Consider f (x) = x2
−10 −5 5 10 −10 −5 5 10
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Functions without Inverses
Not every function has an inverse function, however: Consider f (x) = x2
−10 −5 5 10 −10 −5 5 10
Notice how the inverse is not actually a function - it fails the vertical line test!
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Functions without Inverses
Not every function has an inverse function, however: Consider f (x) = x2
−10 −5 5 10 −10 −5 5 10
Notice how the inverse is not actually a function - it fails the vertical line test! Instead, this is called an inverse relation .
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Functions without Inverses
Not every function has an inverse function, however: Consider f (x) = x2
−10 −5 5 10 −10 −5 5 10
Notice how the inverse is not actually a function - it fails the vertical line test! Instead, this is called an inverse relation . Only one-to-one functions have inverse functions.
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Properties of Inverse Functions
Inverse Function Property (used to show that two functions are inverses): If f and g are inverses of each other, then
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Properties of Inverse Functions
Inverse Function Property (used to show that two functions are inverses): If f and g are inverses of each other, then
(f ◦ g)(x) = x, and
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Properties of Inverse Functions
Inverse Function Property (used to show that two functions are inverses): If f and g are inverses of each other, then
(f ◦ g)(x) = x, and (g ◦ f )(x) = x.
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Properties of Inverse Functions
Inverse Function Property (used to show that two functions are inverses): If f and g are inverses of each other, then
(f ◦ g)(x) = x, and (g ◦ f )(x) = x.
If f and g are inverses, then their domain and range are switched:
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Properties of Inverse Functions
Inverse Function Property (used to show that two functions are inverses): If f and g are inverses of each other, then
(f ◦ g)(x) = x, and (g ◦ f )(x) = x.
If f and g are inverses, then their domain and range are switched:
Domain of f = Range of g
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Properties of Inverse Functions
Inverse Function Property (used to show that two functions are inverses): If f and g are inverses of each other, then
(f ◦ g)(x) = x, and (g ◦ f )(x) = x.
If f and g are inverses, then their domain and range are switched:
Domain of f = Range of g Range of f = Domain of g
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Example
Show that f and g are inverses of each other: f (x) = 2x + 1 and g(x) = 1
2x − 1 2.
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Finding the Formula for an Inverse Function
- 1. Switch x and y (if the function is written as f (x), treat that like
“y”).
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Finding the Formula for an Inverse Function
- 1. Switch x and y (if the function is written as f (x), treat that like
“y”).
- 2. Solve for y.
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Finding the Formula for an Inverse Function
- 1. Switch x and y (if the function is written as f (x), treat that like
“y”).
- 2. Solve for y.
- 3. If necessary, go back to function notation - “y” becomes f −1(x).
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Examples
Find the inverse of each function.
- 1. f (x) = 2x + 4
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Examples
Find the inverse of each function.
- 1. f (x) = 2x + 4
f −1(x) = 1
2x − 2
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Examples
Find the inverse of each function.
- 1. f (x) = 2x + 4
f −1(x) = 1
2x − 2
- 2. f (x) = x3 + 3
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Examples
Find the inverse of each function.
- 1. f (x) = 2x + 4
f −1(x) = 1
2x − 2
- 2. f (x) = x3 + 3
f −1(x) =
3
√x − 3
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Examples
Find the inverse of each function.
- 1. f (x) = 2x + 4
f −1(x) = 1
2x − 2
- 2. f (x) = x3 + 3
f −1(x) =
3
√x − 3
- 3. f (x) = 2x−1
x+3
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Examples
Find the inverse of each function.
- 1. f (x) = 2x + 4
f −1(x) = 1
2x − 2
- 2. f (x) = x3 + 3
f −1(x) =
3
√x − 3
- 3. f (x) = 2x−1
x+3
f −1(x) = 1+3x
2−x
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Examples
Find the inverse of each function.
- 1. f (x) = 2x + 4
f −1(x) = 1
2x − 2
- 2. f (x) = x3 + 3
f −1(x) =
3
√x − 3
- 3. f (x) = 2x−1
x+3
f −1(x) = 1+3x
2−x
- 4. f (x) = √x + 1
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Examples
Find the inverse of each function.
- 1. f (x) = 2x + 4
f −1(x) = 1
2x − 2
- 2. f (x) = x3 + 3
f −1(x) =
3
√x − 3
- 3. f (x) = 2x−1
x+3
f −1(x) = 1+3x
2−x
- 4. f (x) = √x + 1
f −1(x) = (x − 1)2, where x ≥ 1