Section 1: Functions

★ Function : exists if for every input (x-value) there is exactly one output (y-value) * X-value cannot repeat*

★ Domain : x-value input, independent value; ★ Range : y-value output, dependent value★ Function Notation: y= f(x)★ Calculus Notation: Example: x is any number greater than 10: shown below

○ Set builder: {x/x>10; x }∈ℝ○ Interval: (10, ∞)

Questions:1. State the domain for a set of numbers that includes all numbers greater than or equal to

6 using set builder notation and interval notation. *there is a difference between (round brackets) and [square brackets]*

2. Write the following inequality in interval notation: -3 < x ≤ 13

3. Decide whether the following sets of numbers is a function: *Reminder* X-value cannot repeat!!

a)

X Y

3 5

4 9

5 13

6 13

7 9

b)

X Y

5 13

6 17

7 21

7 25

8 30

4. Find the function value. g(x)= 3x 3 ; g(5x) x2 + x -4

5. State the domain of each function: *Domain of different equations: -Line: y=mx+b / ax+by=c; all real numbers -Parabola: y=ax2+b / a(x-h)2+k; all real numbers - Square roots: x≥0; set the radicand ≥ then solve - Fractions: x≠0; set denominator =0, then solve

a. f(x)= 2+x x2-7x

b. f(x)=√4x-1

Answers and solutions:1. Set-builder: {x|x≥6; x }∈ℝ

Interval: [6, ∞)

2. (-3,13]3. a) yes

b) no, x repeats4. g(5x)= 3(5x) 3

(5x)2+5x-4 =5(125x 3 ) 25x2+5x-4 = 375x 3 25x2+5x-4

5. a) x2-7x≠0 x(x-7)≠0 x≠0 x≠7 (-∞,0) (0,7) (7, ∞)∪ ∪ b) 4x-1≥0 4x≥1 x ≥ ¼ [¼, ∞)

Section 2: Analyzing graphs of Functions and Relations

★ Zeroes : where the graph crosses the x-axis○ To find zeros algebraically set equation =0

★ Y-Intercept: where the graph crosses the y-axis○ To find y-intercept set x=0

★ Symmetry ○ X-axis symmetry test: replace (y) with (-y) to produce an equivalent equation

(x,y)→(x,-y) * Not a function; won’t pass vertical line test*○ Y-axis symmetry test/ EVEN Function : replace (x) with (-x) to produce an

equivalent equation (x,y)→(-x,y)○ Origin symmetry/ ODD Function: replace both (x) and (y)with (-x) and (-y) to

produce an equivalent equation (x,y)→(-x,-y)

Questions:

1. State the domain and range of the blue line.

2. Decide whether each equation is even or odd.a. f(x)= x3-2x

b. g(x)= x4+2

3. Find the zeros and y-intercept for the following equation: f(x)= x2-6x+9

Answers: 1. Domain: [-3,3] Range: [5,14]2. a. f(-x)=(-x)3-2(-x)

= -x3+2x = -(x3-2x) = -f(x)

f(-x)=-f(x); function is odd b. g(-x)=(-x)4+2 =x4+2 = g(x) Function is even

3. f(0)=02-6(0)+9 = 0-0+9 = 9 zeros: (0,9)

0=x2-6x+9 = (x-3)(x-3) 3=x y-intercept: (3,0)

Chapter 1 Section 3

Definitions:★ Continuous Function- has no breaks, holes or gaps.★ Limit- concept of approaching a value without necessarily ever reaching it.★ Discontinuous Functions- functions that are not continuous.★ End Behavior- describes how a function behaves at either end of the graph.

lim f(x) lim f(x)x→∞ x→-∞

Types of Discontinuity:

Infinite Discontinuity- the function value increases or decreases indefinitely from left to right.

Jump Discontinuity- the limits of the function from the left and the right exist but have two distinct values.

Removable Discontinuity- if the function is continuous except for a hole at one point.

*Jump and infinite discontinuities are classified as nonremovable discontinuities.*

Questions:

Determine whether each function is continuous at the given x-values. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump or removable.

1. f(x)=1; x=4 x²-16

2. f(x)=x²-36 ;x=-6 x=6 x+6

3. f(x)= {x²-3 if x<0 for x=2}{x+4 if x≥0 }

4.f(x) = {4x-1 if x≤-6 for x=-6}{-x+2 if x>6 }

Describe the end behavior:5.

Use the graph to describe its end behavior . Use a table.6.f(x)=-5x³+7x-1

7.f(x)=12x³+4x -5 (4x³-9

Using a graphing calculator graph each function and determine whether it is continuous. If discontinuous, identify any points of discontinuity. Then describe its end behavior and locate any zeros.

8.x²x³-4x²+x+69.x³-5x²-26x+120 x²+x-12

Answers:1.f(4)=1 = 1 infinite discontinuity (4²)-16 0 2.f(6)=6)²-3)=0 = 0

6+6 12 f(-6)=(-6)²-36)=0= undefined; removable discontinuity at -6

-6+6 03.f(0)=0²-3=-3 x jump discontinuity f(0)=0+4=4 4.f(x)=4(-6)-1=-25 x jump discontinuity f(x)=-(-6 )+2-=85.f(x) =∞ f(x)= =-∞

χ→-∞ χ→∞

6.

x -3 -2 -1 0 1 2 3

y 113 25 -3 -1 1 -27 -115

end behavior: f(x) =∞ f(x) =∞ χ→-∞ χ→∞

7.

x -2 -1 0 1 2

y -10141

-4913

-5 415

-1123

f(x) =-∞ f(x) =∞ χ→-2 χ→-2

8.

discontinuity; x=-1 x=2 x=3 zeros x=0

end behavior: lim f(x)=0 lim f(x)=0 χ→-∞ χ→∞

9.

discontinuity: x=3 x=-4zeros: x=-5 x=4 x=6end behavior:lim g(x)=-∞

χ→-∞lim g(x)=∞ χ→∞

Chapter 1 Section 5★ Parent Function- is the simplest of the functions in a family.

Functions:

Zero function- horizontal line; x is any real number

Identity Function- passes through all points

Quadratic function-f(x)= x²;u-shaped

Cubic function- f(x)=x³; symmetric to the origin

Square root function- f(x)=√x

Reciprocal function- f(x)= 1x

Absolute value function- f(x)=|x|; v-shaped

Greatest integer function- f(x)=[[x]]; greatest integer less than or equal to x.

★ Translations: a rigid transformation that has the effect of shifting the graph of a function.

Vertical translations:

Horizontal translations:

Reflection- rigid transformation that produces a mirror image of the graph of a function with respect to a specific line.

Reflection in the x-axis Reflection in the x-axis

★ Dilations- a nonrigid transformation that has the effect of compressing or expanding the graph of a function vertically or horizontally.

Vertical dilation:

Questions:Describe the following characteristics of the parent function:

Domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.

1.

Use the graph of f(x)=√x to graph the function.2.g(x)=(√x+8)-6

Describe how the graphs are related and then write an equation for g(x).

3.

4.

Identify the parent function f(x) of g(x) and describe how the graph of g(x)and f(x)are related. Then graph f(x)and g(x) on the same axes.5.g(x)=(√+5) 3

6.g(x)=-2|x+5|

Answers:1.D:(-∞,∞) continuity: continuousR:(-∞,∞)Intercept:(0,0)Symmetry: symmetric to the originlim f(x)=-∞ lim f(x)=∞ x→-∞ x→∞2.

3. g(x) is the graph of f(x) translated 4 units down.g(x)=|x|-4

4. g(x) is the graph of f(x) translated 3 units down.g(x)=[[x]]-3

5.g(x) has been translated 5 units to the left and its compressed vertically.

Ch 1 section 6 Operation with functions

Sum (f+g)(x)=f(x) + g(x) Product (f *g)(x)=f(x) * g(x) ( F) f(x)

difference (f--g)(x)= f(x) -- g(x) Quotient( __)(x)= __ , g(x)=/= 0 ( G) g(x)

example 1 operations with functions find each function and its domain f= x2 + 4x g= 3x--5 (f + g)(x)= f(x) + g(x) (f--g)(x)=f(x)--g(x)

(x2 + 4x) + (3x--5) (x2 + 4x) -- (3x --5)combine like terms combinex2+7x--5 x2--x+5

domain of f is (- , ) domains of both is (- , )

and domain g (5, )

so the domain of (5, )(g)

(f*g)(x)=f(x)*g(x) (_)(x)(F)

(x2 + 4x)(3x--5) (3x--5)3x3 -- 5x2 + 12x2 --20x ______

(x2+4x)3x3 + 7x2 -- 20x

domain (-- , ) domain (- ,-4) U (-4,0) U (0, )

question find find each operation and domain for f=x2 + x g=9x

Answer

x2 + x + 9x= x2 + 10x d=(- , ) x2+x--9x=x2--8x d=(- , )

(f+g)x= x2 + 10x d=(- , ) (F--G)x=x2--8x d=(- , )

(x2+ x)(9x)=9x3+9x2 d=(- , ) (x2+x)/(9x)= x + 1/9

d= (- ,0) U(0, )

(f*g)x=9x3+9x2 d=(- , ) (f/g)x= x + 1/9 d= (- ,0) U (0, )

Compositions of functions

f=x2 + 1 g=x--4

find [FoG](x) = F[g(x)] what you do is plug x--4 into the x in x2 + 1 (x--4)2 + 1= x2 --8x+16+1= x2--8x+17

find [GoF](x)=g[F(x)] what do here is same thing as FoG

g(x2+1)--4x2 + 1 --4= x2--3

[Fog](2) what do si u plug equation into each other then you plug 2 into x

(x--4)2 + 1=x2--8x+17=(2)2 --8(2)+17= 5

questions find FoG(x), GoF(x), FoG(6)F(x)= 2x--3G(x)= 4x--8

Answers2(4x --8) --3= 8x--16--3=8x--19 4(2x--3)--8=8x --12--8=8x--20FoG= 8x--19 GoF=8x--20

8(6)--19= 48--19=29FoG(6)=29

Section 7: Inverse Relations and Functions

Inverse relations- exist if and only if one relations contains (b,a) whenever the other relation contains (a,b)

Take notice these inverse relation are reflections of each other in the line y=x. This is true for the graphs of all relations and their inverse. If the inverse relation of a function f is also a function, then it is called the inverse function of f and is denoted f-1 read f inverse.

to tell if there is an inverse function on a graph you must apply the horizontal line test which is where draw a line on a graph to to see if touches another point. it fails if touches more than one point but if it passes then it called a one to one function

to find it algebraically x--1 F(x)=___ replace f(x) with y x+2

x--1F(x)=___ interchange x and y x+2

xy+ 2x= y --1 multiply each side by 2 + y then apply the distributive property

xy-- y= --2x --1 isolate y terms

y(x--1)=--2x--1 distributive --2x--1y=________ solve for y x--1

--2x--1f^--1(x)=________ replace y with F^--1(x) and note that x cannot = 1

x--1

another example

F(X)= x--4 replace f(x) with y

y=x--4 then replace y with F^--1(x)

F^--1(x)=x--4 then that it

verify inverse functions show that F(x)= 6 g(x)= 6

______ ___ + 4x---4 x

show f[g(x)] = x and g[f(x)]

f[g(x)]= f (6) F[g(x)]= g(6) (__ +4) (____ +4) (x) (x--4)

6 6 _________ ________+4 (6) (6) (__ +4)--4 (____) ( x) (x--4)

6 6

_________=x ________+4=x --4+4=x (6) (6) (__) (____) ( x) (x--4)to verify it both functions are inverses both must have same outcome

Find inverse Function Graphically

graph the line y=x locate a few points on the graph of F(x) reflect these points in y=x then connect them with a smooth curve that mirrors the curvature of F(x) in line y=x

questions 1 does it have inverse x2 +6x+9

2 4--x _____ does it have inverse any and any restrictions x3 f(x)=--6x + 3 G(x) 3--x/6 are these two inverses

answers

1 no2yes F-1(x)=4/x+1 x=/= -13--6(3--x)/6 + 3= --18+6x/6 +3= --3 +3+x = x