theorems

Question 1

function

Answer 1

A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The
set of inputs is called the domain of the function. The set of outputs is called the range of the function.

Question 2

vertical line test

Answer 2

Given a function f , every vertical line that may be drawn intersects the graph of f no more than once. If any vertical
line intersects a set of points more than once, the set of points does not represent a function.

Question 3

when we have a graph that’s symmetric with respect to the y axis, we would call it

Answer 3

even

Question 4

x -intercept

Answer 4

let y be zero and solve equation for x

Question 5

y-intercept

Answer 5

let x be zero and solve equation for y

Question 6

symmetry of a graph

Answer 6

There three types of symmetry

1. symmetric with respect to the y-axis. this mean the portion of the graph in the y-axis is the mirror image of the portion of the graph in the y-axis
2. symmetric with respect to the x-axis this mean the portion of the graph in the x-axis is the mirror image of the portion of the graph in the x axis
3. symmetric with respect to the origin

Question 7

Symmetry theorem

Answer 7

1. The graph of an equation in x and y is symmetric with respect to the y-axis when replacing x by -x yields an equivalent equation
2. The graph of an equation in x and y is symmetric with respect to the x-axis when replacing y by -y yields an equivalent equation
3. The graph of an equation in x and y is symmetric with respect to the origin when replacing x by -y and y by -y yields and equivalent eqation .

symmetric to the y test - just change the x to -x
symmetric to the x test - just change the y to -y, which makes the whole function negative

Question 8

The graph of a polynomial has symmetry with respect to the y-axis when each term has

Answer 8

even exponent (constant) y=2x^4-x^2+2

Question 9

the graph of a polynomia has a symmetry with respect to the origin when each term has an

Answer 9

odd exponent

Question 10

even function test

Answer 10

f(-x)=f(x), the graph of cos and sec

Question 11

odd function test

Answer 11

f(-x)=-f(x) the graph of sin, tan, cot, and csc

Question 12

inverse function test

Answer 12

f(g(x))= g(f(x)) = x, they both had yeild the same value

Question 13

vertical line test

Answer 13

Given a function f , every vertical line that may be drawn intersects the graph of f no more than once. If any vertical
line intersects a set of points more than once, the set of points does not represent a function.
if a graph does not pass a vertical line test, then that graph is not a function

Question 14

to find the zeros of a function

Answer 14

just make f(x) equal to zero and solve for the x
for example:
find the zeros of f(x)=x^2+2
x=(x)^1/2

Question 15

three scenarios limit does not exist

Answer 15

• Asymptotes
• break and jump
• biesce wise function

Question 16

if a function is differentiabile

Answer 16

continuous

Question 17

if a function is not differentiable

Answer 17

not continuous and you can not find the derivative

Question 18

three scenarios that derivative, does not exist

Answer 18

vertical line, for example: (x)^1/3

sharp point: |x|

Question 19

squeeze theorem

Answer 19

If f, g, and h are functions defined on some open interval containing a such that g(x)<=f(x),=h(x) for all x in the interval except possibly at a itself, and 
lim g(x) =    lim h(x)   = L  then the lim f(x)    = L
x--> a           x-->a                           x--->

Question 20

infinite limits

Answer 20

if f is a function defined at every number in some open interval containing a, except possible at a itself, then
if a function approaches to the positive infinity, we’re going to say f(x) increases without bound as x approaches a.
if a function approaches to the negative infinity, we’re going to say f(x) decreases without bound as a x approaches a.

Question 21

limit theorems

Answer 21

1. If n is a positive integer, then

Question 22

the limit of a function as x approaches to infinity, the power of the numerator is less than the power of the denominator, so the limit is

Answer 22

0

Question 23

if denominator is equal with numerator, the limit to infinity is equal to

Answer 23

coefficient

Question 24

if the degree of denominator is less than numerator, just evaluate the limit by

Answer 24

reducing the variable and find for the real limit

Question 25

the limit of a constant

Answer 25

constant

Question 26

finding the aymptote using limit

Answer 26

just make it approach to infinity then u’re going to find horizontal asymptote
just make it approach to the denominator and u’re going to find vertical asymptote

Question 27

a function said to be continuous

Answer 27

f(a) exists
lim    f(x) =   exists
x---> a
lim    f(x) =   f(a)
x---> a

Question 28

continuous

Answer 28

a graph with no hole or break
continuity theorem:
1. A polynomial function is continuous every where
2. A rational function is continuous, except at a point results the denominator to zero
3. If a function f and g are continuous at a, then the functions f+g, f-g, f*g, f/g, and g(a)not equal to 0, are also continuous at a.

Question 29

Intermediate value theorem

Answer 29

If a function f is continuous on a closed interval [a, b] and k is a number with f(a) <=k<=f(b), then there exists a number c in [a, b] such that f(c) =k

Question 30

differentiability

Answer 30

given a function f, if f’(x) exists at x=a, then the function f is said to be differentiable at x=a. If a function is differentiable at x=a, the f is continuous at x=a.
If a function f is continuous at x=a, then f may or may not be differentiable at x=a.
cases for differentiability
cusp = not differentiable
vertical tangent = not differential
has corner= not differential

Question 31

to know if a function has an inverse

Answer 31

just find the derivative of the function and compare with zero or check if it is greater than zero and the function would be increasing and it would pass the horizontal line test.
if a function passes a horizontal line test and that function has an inverse

Question 32

higher order derivatives

Answer 32

if the derivative f’ of a function f is differentiable, then the derivative of f’ is the second derivative of f represented by f’’ (reads as f double prime). You can continue to differentiate f as long as there is differentiability.

Question 33

Rolle’s Theorem

Answer 33

if f is continuous on a closed interval [a, b], f is differentiable on the open interval (a,b) and f(a)=f(b)=0, then there exists a number c where f’(c)=0

Question 34

Mean value Theorem

Answer 34

if a f is continuous on a closed interval [a, b] and f is differentiable on the open interval (a, b) then there exists a number c in (a, b) such that f’(c) =f(b)-f(a)/b-a