1. R R { [ -8, ) } R { [ 0, ) } { [ 4, ) } { [ 0, ) } { (- , 3 ] }
{ (- , 4 ] U [ 2, ) }
{ (- 3, ) }
{ (- , -1) U [ 0, ) }
{ [ 0, ) }
Pre-Calculus

2. discontinuous - infinite discontinuous - infinite
removable
continuous
discontinuous
removable
discontinuous
jump
discontinuous - jump
continuous
continuous
continuous
discontinuous - infinite
discontinuous - infinite
Pre-Calculus

3. (3x+4)(x<1)+(x-1)(x>1) (3+x2)(x<-2)+(2x)(x>-2)
jump
(x^3+1)(x0)+
(2)(x=0)
removable
(3+x2)(x<-2)+(2x)(x>-2)
(x<1)+(11-x2)(x>1)
jump
Pre-Calculus

4. incr: (- , )
decr: (- , 0 ]
incr: [ 0, )
decr: (- , 0 ]
incr: [ 0, )
decr: [ - 1, 1 ]
incr: (- , -1 ], [ 1, )
decr: [ 3, 5 ], incr: [ , 3 ]
constant: [ 5, )
decr: [ 3, ), incr: ( 0 ]
constant: [ 0, 3)
decr: ( - ,  )
decr: ( 0,  )
incr: ( - , 0 )
decr: (- , -8 ]
incr: [ 8,  )
decr: ( 2,  )
incr: ( - , 2)
constant: [ -2, 2 ]
decr: ( - , 0 ]
incr: [ 0, 3 )
constant: [ 3,  )
decr: ( - , 7 )
decr: ( 7,  )
Pre-Calculus

5. bounded below unbounded bounded below b = 0 b = 1 bounded above
Left branch:
bounded above
B = 5
unbounded
bounded above
B = 0
bounded
b= -1, B = 1
Right branch:
bounded below
b = 5
bounded below
b = 0
bounded below
b = -1
bounded below
b = 0
bounded above
B = 0
Pre-Calculus

6. The graph looks the same to the
y-axis
EVEN functions
The graph looks the same to the
left of the y-axis as it does to the right
For all x in the domain of f,
f(-x) = f(x)
x-axis
The graph looks the same above
the x-axis as it does below it
(x, - y) is on the graph whenever
(x, y) is on the graph
origin
ODD functions
The graph looks the same upside
Down as it does right side up
For all x in the domain of f,
f(-x) = - f(x)
Pre-Calculus

7. Odd Even Even Odd Even Neither Even Neither Even Odd
Pre-Calculus

8. horizontally vertically will not cross asymptotes tan and cot x = -1
End behavior
Limit notation
Pre-Calculus

9. Horizontal: y = 0 Vertical: x = 2, -2 Horizontal: y = 0
Pre-Calculus

10. Each x-value has only 1 y-value
Yes
Each x-value has only 1 y-value
{ ( - , -1 ) U (-1, 1) U (1,  ) }
{ ( - , 0) U [ 3,  ) }
Infinite discontinuity
Decreasing: (- , -1), (-1, 0 ]
Increasing: ([ 0, 1), (1,  )
Unbounded
Left piece: B = 0, Middle piece b = 3, Right piece B = 0
Local min at (0, 3)
Even
Horizontal: y = 0, Vertical: x = -1, 1
Pre-Calculus

11. Each x-value has only 1 y-value
Yes
Each x-value has only 1 y-value
{ ( - ,  ) }
{ [ 0,  ) }
continuous
Decreasing: (- , 0 ]
Increasing: [ 0,  )
Bounded below b = 0
Absolute min = 0 at x= 0
Neither even or odd
none
{ ( - , -3 ] U [ 7,  ) }
Pre-Calculus

12. 10 Basic Functions
Pre-Calculus

13. In-class Exercise Section 1.3 Domain Range Continuity Increasing
Decreasing
Boundedness
Extrema
Symmetry
Asymptotes
End Behavior
Pre-Calculus

14. f(x)/g(x), provided g(x)  0
3x3 + x2 + 6
3x3 – x2 + 8
3x5 – 3x3 + 7x2 – 7
x2 – (x + 4) = x2 – x – 4
Pre-Calculus

15. sin(x) x2 +, –, x,  applying them in order the squaring function
the sin function
function composition
f ○ g
(f ◦ g)(x) = f(g(x))
4x2 – 12x + 9 1
2x2 – 3 5 x4
4x – 9
Pre-Calculus

16. Pre-Calculus

17. both vertical and horizontal
inverse
functions
horizontal line test
original relation
Graph is a function (passes vertical line test.
Inverse is also a function (passes horizontal line test.)
Graph is a function (passes vertical line test.
Inverse is not a function (fails horizontal line test.)
both vertical and horizontal
line test like A
one-to-one function
is paired with a unique y
is paired with a unique x
inverse function
f –1
f –1 (b) = a, iff f(a) = b
Pre-Calculus

18. Pre-Calculus

19. D: { ( - ,  ) } R: { ( - ,  ) } D: { ( - ,  ) }
D: { ( - , - 2) U ( -2,  ) }
R: { ( - , 1) U (1,  ) }
D: { ( - , 1) U (1,  ) }
Pre-Calculus

20. inside function outside function x2 + 1 x2
Pre-Calculus

21. f(x) and g(x) are inverses
{ ( - ,  ) }
{ ( - ,  ) }
{ ( - ,  ) }
{ ( - , - 5) U
( - 5, ) }
{ ( - ,  ) }
{ ( - ,  ) }
f(x) and g(x) are inverses
Pre-Calculus

22. passes horizontal line test
Yes
passes horizontal line test
Yes
D: { ( - , 0 ) U ( 0,  ) }
R: { ( - , 4 ) U ( 4,  ) }
D: { ( - , 4 ) U ( 4,  ) }
Pre-Calculus

23. D: { ( - , - 2 ) U ( - 2, 1 ) U ( 1,  ) }
D: { ( - , 2/3 ) U ( 2/3, 1 ) U ( 1,  ) }
D: { (- , - 2) U (- 2, 1) U ( 1,  ) }
D: { ( - , 0 ) U ( 0,  ) }
Pre-Calculus

24. add or subtract a constant to the entire function
f(x) + c
up c units
f(x) – c
down c units
add or subtract a constant to x within the function
f(x – c)
right c units
f(x + c)
left c units
Pre-Calculus

25. negate the entire function y = – f(x)
reflections
negate the entire function y = – f(x)
negate x within the function y = f(-x)
Pre-Calculus

26. multiply c to the entire function
Stretch if c > 1
Shrink if c < 1
multiply c to x within the function
Stretch if c > 1
Shrink if c < 1
A reflection combined
with a distortion
complete any stretches, shrinks or reflections first
complete any shifts (translations)
Pre-Calculus

27. Answers y = 1/x 4 y = x, y = x3, y = 1/x, y = ln (x) y = sqrt(x)
y = 2sin(0.5x)
Stretch by 8
Shrink ½
Shrink by 1/8
Stretch by 2
Pre-Calculus