math-3 lesson 4-1 inverse functions. definition a function is a set of ordered pairs with no two...
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Math-3
Lesson 4-1
Inverse Functions
Definition
• A function is a set of ordered pairs with no two first elements alike. – f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) }
• But ... what if we reverse the order of the pairs?– This is also a function ... it is the inverse function – f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }
Does This Have An Inverse?
• Given the function at the right– Can it have an inverse?– Why or Why Not?
• NO … when we reverse the ordered pairs, the result is Not a function– We would say the function is
not one-to-one
• A function is one-to-onewhen different inputs always result in different outputs
x Y
1 5
2 9
4 6
7 5( ) ( )c d f c f d
Is the Inverse of a function a function?
-2
-1
0
1
2
3
4
5
6
0 2 4 6 8 10
Horizontal line test: if a horizontal line passesthrough the graph of the relation at more than onelocation, then the inverse of that relation is NOTa function.
0
1
2
3
4
5
6
7
8
9
10
-2 -1 0 1 2 3 4 5 6
2)2( xy
Fails horizontal line test
xy 22)2( yx or
The inverse relation is not a function.
One-to-One functions
0
1
2
3
4
5
6
7
8
9
10
-2 -1 0 1 2 3 4 5 6 -1.5
-1
-0.5
0
0.5
1
1.5
0 50 100 150 200 250 300 350 400
-8
-6
-4
-2
0
2
4
6
8
-3 -2 -1 0 1 2 3 4 5
For every input there is exactly one output (the definition of a function) AND every output has exactly one input. More simply:
a. b. c.
Which function is “one-to-one” ?
It passes both the horizontal and vertical line test.
One-to-One Functions
Each input has exactly one output and each output has exactly one input.
If the function passes the Horizontal line test, then its inverse is also a function.
3( )f x x
2( )f x x
Domain and Range
• The domain of f is the range of f -1 • The range of f is the domain of f -1
• Thus ... we may be required to restrict the
domain of f so that f -1 is a function
Restricting the domain of f(x) so that its inverse is a function.Cut the function into pieces using a vertical line so that it passes the horizontal line test.
2)2()( xxf
For x ≥ 2, you get the right ½ of the parabola which passes the horizontal line test.
})2( ,2:),{()( 2 xyxyxxg
Rewrite the function so the it applies for only x-values x ≥ 2
xxg 2)(1
g(x) and it’s inverse are inverses of each other.
)(1 xg
Finding the Inverse: exchange the locations of ‘x’ and ‘y’ in the equation then solve for
‘y’.2)2()( xxf
2)2( xy2)2( yx
2)2( yx
2yx
yx 2
xxy
Domain and RangeConsider the function h(x) = x2 – 9Determine the inverse function
Problem => f -1(x) is not a function
Rewrite the function so the it applies for only x-values x ≥ 0
}9 ,0:),{()( 2 xyxyxxg
}9 :),{()(1 xyyxxg
Inverse RelationsInverse the two (4, 2)(4, 2)(x, y) = (2, 4)
What is the pattern?
A reflection acrossthe line y = x.
(1, -3)(1, -3)Inverse the two(x, y) = (-3, 1)
Inverse Reflection Principle
Inverse Functions12 xy 12 yx
2
1
2x
y
We’re not used to graphing‘y’ as an input value, thenfinding the output value ‘x’
So…we can rewrite the equation as ‘y’ in terms of ‘x’(it’s the same relation).
Switch ‘x’ and ‘y’
Bottom line: inverse functions are reflections across the line y = x.
Is the Inverse Relation a Function?x y
1 2
2 4
3 2
4 1
x y
2 1
4 2
2 3
1 4
Your Turn:3. 3. Draw the following graph of: Is the inverse relation a function?
4. 4. On the same x-y plot draw
3)( xxf
)(1 xf
)(1 xf
Inverse Function Defined
abf )(1 baf )(
If f(x) is a one-to-one function with Domain “D” and Range “R” then the inverse function of f(x), denoted
)(1 xf
Is a function whose Domain is “R” and whose RangeIs “D” defined by:
if and only if
This is just saying the domain of a function is the range of its inverse function.
Natural Logarithm Function
xxf ln)(
Exponential FunctionExponential Function
xexf )(
Domain = ?
Range = ?
Domain = ?
Range = ?
Finding the Inverse function algebraiciallyFinding the Inverse function algebraicially
?)(1 xf
Write the function in “y = “ format.
63)( xxf
63 xy
Exhange ‘x’ and ‘y’ in the equation: 63 yx
Solve for ‘y’ to find the inverse function:3
6x
y
23
1 xy
Finding Inverse Functions Algebraically
-18
-15
-12
-9
-6
-3
0
3
6
9
12
15
18
-4 -3 -2 -1 0 1 2
1)(
x
xxf
Is this function “one-to-one”?
1. Rewrite the function so that ‘y’ is in terms of ‘x’. 1
x
xy
3. Exchange ‘x’ and ‘y’ in the equation. 1
y
yx
Your turn: 5.Your turn: 5. Solve for ‘y’.
x ≠ -1
2. See if the inverse functionexists by checking if f(x) is one-to-one. State any restrictionson the domain of f(x).
x
xy
1
1y
yx
-5
-4
-3
-2
-1
0
1
2
3
-20 -15 -10 -5 0 5 10 15 20
Solve for ‘y’.
yyx )1(
yxxy xyxy xyxy
xxy )1(
1
x
xy
Finding inverse functions graphically can be easierUsing the inverse reflection principle.
Domain: x ≠ 1
1)(
x
xxf
x
xy
1
Finding Inverse Functions (again)
0
0.5
1
1.5
2
2.5
3
3.5
4
-4 -2 0 2 4 6 8 10 12
1. Rewrite the function so that ‘y’ is in terms of ‘x’.
2. See if the inverse function exists by checking if f(x) is one-to-one. State any restrictions on the domain of f(x) to ensure that it is one-to-one.
3)( xxf
3 xy
Passes horizontal Passes horizontal line test.line test.
y ≥ 0y ≥ 0
x ≥ -3x ≥ -3
y ≥ 0y ≥ 0
x ≥ -3x ≥ -3
The domain of f(x) is the range of the inverse function. The range of f(x) is the domain
of the inverse function.
Finding Inverse Functions (again)
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
3. Switch the location of ‘x’ and ‘y’.
4. Solve for ‘y’.
3 yx
22 3 yx 32 xyy ≥ -3 x ≥ 0
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
Inverse functions:3)( xxf 31 )( xxf
} ,0:),{()( 2xyxyxxf xxf )(1
xxf log)( xxf 10)(1
Completed to here on 11-11-14.
Verifying Inverse Functions AlgebraicallyIF f(g(x)) = x (for every ‘x’ in the domain of g(x))
And
IF g(f(x)) = x (for every ‘x’ in the domain of f(x)
THEN: f(x) is a one-to-one function with inverse g(x)
Verifying Inverse Functions
1)( 3 xxf 3 1)( xxg
11))((3
3 xxgf xx 11
3 3 1)1())(( xxfg xx 3 3 11
Your Turn:6. 6. Verify that the two functions are inverses of each other.
x
xxf
1)(
1
1)(
x
xg
7. 7. Verify that the two functions are inverses of each other.
3
2)(
x
xf 23)( xxg