InversesMA2A2. Students will explore inverses of functions.
Discuss the characteristics of functions and their inverses, including one-to-oneness, domain, and range.Determine inverses of linear, quadratic, and power functions and functions of the form , including the use of restricted
domains.Explore the graphs of functions and their inverses.
Use composition to verify that functions are inverses of each other.
A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain.
A function is not one-to-one if two different elements in the domain correspond to the same element in the range.
Theorem Horizontal Line Test
If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.
Domain of f Range of f
Range of f 1 Domain of f 1
f 1
f
Domain of Range of
Range of Domain of
f f
f f
1
1
Theorem
The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.
f 1
Finding the inverse of a 1-1 functionStep1: Write the equation in the form
Step2: Interchange x and y.
Step 3: Solve for y.
Step 4: Write for y.
)(xfy
)(1 xf
Even and Odd Functions
MA2A3. Students will analyze graphs of polynomial functions of higher degree. b. Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of
real zeros.c. Determine whether a polynomial function has symmetry and whether it is even, odd, or neither.
d. Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior.
Even functionsA function f is an even function if
for all values of x in the domain of f.
Example: is even because
)()( xfxf
13)( 2 xxf
)(131)(3)( 22 xfxxxf
Odd functionsA function f is an odd function if
for all values of x in the domain of f.
Example: is odd because
)()( xfxf
xxxf 35)(
)()5(5)(5)( 333 xfxxxxxxxf
Graphs of Even and Odd functions
The graph of an even function is symmetric with respect to the y-axis.
The graph of an odd function is symmetric with respect to the origin.
Synthetic DivisionMA2A3. Students will analyze graphs of polynomial functions of higher degree.
b. Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros.
c. Determine whether a polynomial function has symmetry and whether it is even, odd, or neither. d. Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative
and absolute extrema, intervals of increase and decrease, and end behavior.
3 2
2
3x x 4x 1x 1
2 3 2x 1 3x x 4x 1
3
2
3xx
3x
3x
33x 3x2x x 1
2 3 2x 1 3x x 4x 1
3x
33x 3x2x x 1 2
2 1xx
1
2x 1x
2
x3x 1x 1
4 2
2
x 2x x 3x x 1
2 4 230xx x 1 x 2x x 3
4
22x
xx
4x 3x3 2x x x
2x x
2x
2x
2 4 3 2x x 1 x 0x 2x x 3 4x 3x
3 2x x x
2x
2x
3
2 xxx
x
3x 2x x
3 2x 2 x 0x x 2 3x
22x x 2
2x
22x
3x x 2x 2
3 20x 2 x 2xx
23xx
x
3x22x x 2
2x
22x
22xx
2x
2x
22x 4x5x 25x
x5
5
5x 10
12
2 12x 2x 5x 2
3 22x x 2x 3x 1
3 2x 1 2x x 2x 3 3
22xx
2x
3 22x 2x23x 2x 3
22x
23xx
3x
3x
23x 3x
x 3
1xx
1
x 1 4
2 42x 3x 1x 1
Synthetic Division Summary1. Set denominator = 0 and solve (box number)2. Bring down first number3. Multiply by box number and add until finished4. Remainder goes over divisor
Notes of Caution1. ALL terms must be represented (even if coefficient is 0)2. If box number is a fraction, must divide final answer by the denominator
To evaluate a function at a particular value, you may EITHER:A) Substitute the value and simplify ORB) Complete synthetic division…the remainder is your answer
3 2x 4x 2x 5x 3
xx3
30
1 -4 2 -5 3
1
3
-1
-3
-1
-3
-8
2x x 1x83
3 22x x 2x 3x 1
xx110
2 -1 2 -3 1
2
2
1
1
3
3
0
22x x 3
3 24x 3x 8x 4x 3
xx3
30
4 -3 -8 4 3
4
12
9
27
19
57
61
24x 9x 1 619x 3
3 22x 5x 28x 14x 5
xx5
50
2 -5 -28 14 5
2
10
5
25
-3
-15
-1
22x 5xx135
3 216x 32x 81x 162x 2
xx2
20
16 -32 -81 162 2
16
32
0
0
-81
-162
0
216x 81
3 2x 2x x 1x 3
xx3
30
1 -2 -1 1 3
1
3
1
3
2
6
7
2x x 2x75
3x 5x 2x 3
xx3
30
1 0 -5 2 3
1
3
3
9
4
12
14
2x 3x 4x43
1
4 2x 17x 16x 4
xx4
40
1 0 -17 0 16 4
1
4
4
16
-1
-4
-4
3 2x 4x x 4
23x 5x0 x 2
3x
4 23x 17x 1x 604
0xx
-16
0
3 26x 4x 3x 23x 2
3x 2 023
x
6 -4 3 -2 2/3
6
4
0
0
3
2
0
22x 1
4 3 22x 5x 4x 5x 22x 1
2x 1 01x2
2 5 4 5 2
2
-1
4
-2
2
-1
4
3 2x 2x x 2
-2
0
3
-1/2
2
3
x 3x 5x 2
x 3 0x 3
1 0 -5 2 3
1
3
3
9
4
12
14
2 14x 3x 4x 3
3 2x 0x 5x 2x 3
3Find f 3 if f x x 5x 2
f 3 14
4 2x 17xx 4
16
x 4 0x 4
1 0 -17 0 16 4
1
4
4
16
-1
-4
-4
3 2x 4x x 4
4 3 2x 0x 17x 0x 16x 4
-16
0
4 2Find f 4 if f x x 17x 16
f 4 0
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