Outline Inverse Functions One-to-One Functions
College Algebra & Trigonometry I 2.7 - Inverse Functions Math 1100 North Carolina Central University Math & C.S. Department Hicham Qasmi -
[email protected]
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
1
Inverse Functions Definition Example Find Function Inverses
2
One-to-One Functions Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
1
Inverse Functions Definition Example Find Function Inverses
2
One-to-One Functions Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Outline
1
Inverse Functions Definition Example Find Function Inverses
2
One-to-One Functions Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Inverse Functions Theorem Let f and g be 2 functions so that f g(x) = x for every x in the domain of g and g f (x) = x for every x in the domain of f The function g is called the inverse of f and is denoted by f −1 . Thus, f f −1 (x) = x and f −1 f (x) = x The domain of f −1 is the range of f The range of f −1 is the domain of f .
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Inverse Functions Theorem Let f and g be 2 functions so that f g(x) = x for every x in the domain of g and g f (x) = x for every x in the domain of f The function g is called the inverse of f and is denoted by f −1 . Thus, f f −1 (x) = x and f −1 f (x) = x The domain of f −1 is the range of f The range of f −1 is the domain of f .
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Inverse Functions Theorem Let f and g be 2 functions so that f g(x) = x for every x in the domain of g and g f (x) = x for every x in the domain of f The function g is called the inverse of f and is denoted by f −1 . Thus, f f −1 (x) = x and f −1 f (x) = x The domain of f −1 is the range of f The range of f −1 is the domain of f .
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Inverse Functions Theorem Let f and g be 2 functions so that f g(x) = x for every x in the domain of g and g f (x) = x for every x in the domain of f The function g is called the inverse of f and is denoted by f −1 . Thus, f f −1 (x) = x and f −1 f (x) = x The domain of f −1 is the range of f The range of f −1 is the domain of f .
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Inverse Functions Theorem Let f and g be 2 functions so that f g(x) = x for every x in the domain of g and g f (x) = x for every x in the domain of f The function g is called the inverse of f and is denoted by f −1 . Thus, f f −1 (x) = x and f −1 f (x) = x The domain of f −1 is the range of f The range of f −1 is the domain of f .
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Inverse Functions Theorem Let f and g be 2 functions so that f g(x) = x for every x in the domain of g and g f (x) = x for every x in the domain of f The function g is called the inverse of f and is denoted by f −1 . Thus, f f −1 (x) = x and f −1 f (x) = x The domain of f −1 is the range of f The range of f −1 is the domain of f .
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Outline
1
Inverse Functions Definition Example Find Function Inverses
2
One-to-One Functions Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Example of the Inverse of a Functions
Example Show that each function is the inverse of the other: f (x) = 3x + 2 and g(x) =
Math 1100
x −2 3
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Outline
1
Inverse Functions Definition Example Find Function Inverses
2
One-to-One Functions Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Find the Inverse of a Function 1
Replace f (x) in the equation with y −→ for f (x) = x + 2, it means y = x + 2
2
Interchange x and y in the equation y = f (x) −→ x = y + 2
3
Solve for y −→ y = x − 2
4
If the equation does not define y as a function of x, then the function does not have an inverse. If it does, then f has an inverse function −→ This defines y as a function of x, so f −1 exists
5
If f has an inverse function, replace y in step 3 by f −1 (x) −→ So, f −1 (x) = x − 2 Verify f f −1 (x) = x and f −1 f (x) = x. −→ f f −1 (x) = (x − 2) + 2 = x −→ f −1 f (x) = (x + 2) − 2 = x
6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Definition Example Find Function Inverses
Example Find the inverse of: 1 2 3
f (x) = 7x − 2
g(x) = x 3 + 1 f (x) =
5 x
+4
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Outline
1
Inverse Functions Definition Example Find Function Inverses
2
One-to-One Functions Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Function Without Inverse
Example Try to find the inverse of f (x) = x 2 + 1: f (x) = x 2 + 1 y = x2 + 1 x = y2 + 1 y2 = x √ −1 y =± x −1 Since y is not uniquely defined by x, There is no inverse!
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Function Without Inverse
Example Try to find the inverse of f (x) = x 2 + 1: f (x) = x 2 + 1 y = x2 + 1 x = y2 + 1 y2 = x √ −1 y =± x −1 Since y is not uniquely defined by x, There is no inverse!
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Function Without Inverse
Example Try to find the inverse of f (x) = x 2 + 1: f (x) = x 2 + 1 y = x2 + 1 x = y2 + 1 y2 = x √ −1 y =± x −1 Since y is not uniquely defined by x, There is no inverse!
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Function Without Inverse
Example Try to find the inverse of f (x) = x 2 + 1: f (x) = x 2 + 1 y = x2 + 1 x = y2 + 1 y2 = x √ −1 y =± x −1 Since y is not uniquely defined by x, There is no inverse!
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Function Without Inverse
Example Try to find the inverse of f (x) = x 2 + 1: f (x) = x 2 + 1 y = x2 + 1 x = y2 + 1 y2 = x √ −1 y =± x −1 Since y is not uniquely defined by x, There is no inverse!
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Function Without Inverse
Example Try to find the inverse of f (x) = x 2 + 1: f (x) = x 2 + 1 y = x2 + 1 x = y2 + 1 y2 = x √ −1 y =± x −1 Since y is not uniquely defined by x, There is no inverse!
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Outline
1
Inverse Functions Definition Example Find Function Inverses
2
One-to-One Functions Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Horizontal Line Test
Theorem A function f has an inverse if there is no horizontal line that intersects the graph of y = f (x) at more than one point.
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Which of these functions have inverse functions?
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Which of these functions have inverse functions?
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Which of these functions have inverse functions?
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Which of these functions have inverse functions?
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Outline
1
Inverse Functions Definition Example Find Function Inverses
2
One-to-One Functions Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
One-to-One Function Theorem A function f is a one-to-one function if for any x1 and x2 f (x1 ) = f (x2 )
means x1 = x2
(A One-to-One Function is also called an injection) In other words, no 2 different ordered pairs have the same second y-coordinate. A One-to-One function always verifiy the Horizontal line test Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
One-to-One Function Theorem A function f is a one-to-one function if for any x1 and x2 f (x1 ) = f (x2 )
means x1 = x2
(A One-to-One Function is also called an injection) In other words, no 2 different ordered pairs have the same second y-coordinate. A One-to-One function always verifiy the Horizontal line test Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
One-to-One Function Theorem A function f is a one-to-one function if for any x1 and x2 f (x1 ) = f (x2 )
means x1 = x2
(A One-to-One Function is also called an injection) In other words, no 2 different ordered pairs have the same second y-coordinate. A One-to-One function always verifiy the Horizontal line test Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Outline
1
Inverse Functions Definition Example Find Function Inverses
2
One-to-One Functions Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Graphing the inverse of a Function
Property The graph of f −1 is a reflection of the graph of f about the line y = x (1st quadrant bisector)
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Graphing the inverse of a Function
Property The graph of f −1 is a reflection of the graph of f about the line y = x (1st quadrant bisector)
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Graph of f deduced from the Graph of f −1
6 5 4 3 2 1 0
y = f (x)
0 1 2 3 4 5 6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Graph of f deduced from the Graph of f −1
6 5 4 3 2 1 0
y = f (x) (a, b)•
0 1 2 3 4 5 6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Graph of f deduced from the Graph of f −1
6 5 4 3 2 1 0
y = f (x) y =x (a, b)•
reflection
0 1 2 3 4 5 6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Graph of f deduced from the Graph of f −1
6 5 4 3 2 1 0
y = f (x) y =x (a, b)•
reflection (b, a) •
0 1 2 3 4 5 6
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Graph of f deduced from the Graph of f −1
6 5 4 3 2 1 0
y = f (x) y =x (a, b)•
reflection (b, a) •
y = f −1 (x)
0 1 2 3 4 5 6
Math 1100
College Algebra & Trigonometry I
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Outline Inverse Functions One-to-One Functions
Example of Graph of f −1 Use the graph of f below to draw the graph of f −1 5 4 3
(4, 2)
2
•
y = f (x)
1
(−1, 0)
0 −5 −4 −3 −2 −1 0 −1 •
1
2
3
4
5
(−3, −2) •
−2 −3 −4 −5
Math 1100
College Algebra & Trigonometry I
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Outline Inverse Functions One-to-One Functions
Example of Graph of f −1 Use the graph of f below to draw the graph of f −1 5 4
(2, 4) •
3
(4, 2)
2
•
y = f (x)
1
(−1, 0)
0 −5 −4 −3 −2 −1 0 1 2 • 0) −1(−1, •
3
4
5
(−3, −2) •
−2 •
−3
(−2, −3)
−4 −5
Math 1100
College Algebra & Trigonometry I
Outline Inverse Functions One-to-One Functions
Function Without Inverse Horizontal Line Test Definition Graph of f and f −1
Example of Graph of f −1
1
2
Find the inverse of f (x) = x 2 − 1 when the domain of f is limited to the positive numbers. For x ≤ 0, graph f and f −1 in the same rectangular coordinate system.
Math 1100
College Algebra & Trigonometry I
