Flashcards in Math 150 Chapter 1 - 2.6 Deck (29)

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1

## Definition of function

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A function (or map) is a rule or correspondence that associates each element of a set

X, called the domain, with a unique element of a set Y , called the codomain.

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##
x={Tom,Jerry, and Sam} and y= {3, 4, 5, 6, 7}

If the pairs are (Tom,3), (Jerry, 5) and (Sam, 7), what is the domain, range and codomain?

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Domain: {Tom, Jerry and Sam}

Range : {3, 5, 7}

Codomain: {3,4,5,6,7}

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## Definition of one-to-one function

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one-to-one function if it never takes on the same value twice; that is

if x1 ≠ x2, then f(x1) ≠ f(x2).

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## Definition of inverse function

### Let f be one-to-one function with domain A and range B. Then its inverse function has domain B and range A and is defined by f^-1(y) = x f(x) = y

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## Definition of arcsine or sin^(-1)

### The inverse function of the sine function f(x) = sin x with the domain of -π/2 ≤ x ≤ (π)/2

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## Definition lim┬(x→a)〖f(x)〗= L

### the limit of f(x), as x approaches to a, equals L

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## Definition of lim┬(x→a)〖f(x)〗= ∞

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It means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x

sufficiently close to a, but not equal to a

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## Definition of vertical asymptote for limits

### The line x = a is a vertical asymptote if at least one of the side limits equals infinity or negative infinity. (Ex. lim┬(x→a)〖f(x)〗= ∞)

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## Definition of continuous (2 definitions, same meaning)

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- A function f is continuous at a number a if lim┬(x→a)〖f(x)〗= f(a)

- A function f is continuous from the right at a number a if lim┬(x→a^+ ) [f(x)] =f(a)

and f is continuous from the left at a if

lim┬(x→a^- ) [f(x)]=f(a)

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## Definition of discontinuous

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If

(1) f is defined on an open interval containing a, except perhaps at a, and

(2) f is not continuous at a

we say that f is discontinuous at a.

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## Definition of lim┬(x→∞)f(x) = L (2 things)

### It means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Also this is the definition of a horizontal asymptote.

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## How do you find horizontal asymptotes?

### By taking the limit as x approaches infinity and by also taking the limit as x approaches negative infinity.

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## Definition of horizontal asymptote

### The line y = L is called a horizontal asymptote if lim┬(x→∞)f(x) = L or lim┬(x→ -∞)f(x) = L

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## What is the Factor Theorem

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Let P(x) be a polynomial and r a real number. If P(r) = 0, then x-r is a factor of

P(x), i.e. P(x) = (x-r)Q(x) for some polynomial Q(x). Also, if x-r is a factor of P(x), then P(r) = 0.

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## Explain this Theorem. lim┬(x→a) f(x) ≤ lim┬(x→a) g(x)

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If f(x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both exist

as x approaches a, then lim┬(x→a) f(x) ≤ lim┬(x→a) g(x)

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## What is the Squeeze Theorem?

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If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and lim┬(x→a) f(x) = lim┬(x→a) h(x) = L, then

lim┬(x→a) g(x) = L

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##
Explain this theorem. lim┬(x→a) f(g(x)) =

f(lim┬(x→a) g(x)) = f(b)

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If If f is continuous at b and lim┬(x→a) g(x) = b then

lim┬(x→a) f(g(x)) = f(lim┬(x→a) g(x)) = f(b)

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## Explain this theorem. The composite function f o g given by (f o g)(x) = f(g(x)) is continuous at a.

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If g is continuous at a and f is continuous at g(a), then the composite function f o g given

by (f o g)(x) = f(g(x)) is continuous at a.

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## What is the intermediate value theorem?

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Suppose that f is continuous on the closed interval

[a,b] and let N be any number between f(a) and f(b), where f(a) ≠ f(b). Then there exists a number c in (a, b) such that f(c) = N.

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## What is the first three limit laws? What are the conditions for the limit laws to work?

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Conditions: c is a constant and both the limits lim┬(x→a) f(x) and lim┬(x→a) g(x) EXISTS.

1) lim┬(x→a)[f(x)+g(x)] = lim┬(x→a) f(x)+ lim┬(x→a) g(x)

2) lim┬(x→a)[f(x)-g(x)] = lim┬(x→a) f(x) - lim┬(x→a) g(x)

3) lim┬(x→a) (c(f(x))) = c[lim┬(x→a) f(x)]

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## What is the other three limit laws? What are the conditions for the limit laws to work?

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Conditions: c is a constant and both the limits lim┬(x→a) f(x) and lim┬(x→a) g(x) EXISTS.

4) lim┬(x→a)[f(x)g(x)] = [lim┬(x→a) f(x)] [lim┬(x→a) g(x)]

5) lim┬(x→a)[f(x)÷g(x)] = [lim┬(x→a) f(x)]÷[lim┬(x→a) g(x)], if lim┬(x→a) g(x) ≠ 0

6) lim┬(x→a) [f(x)]^(p/q) = [lim┬(x→a) f(x)]^(p/q)

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## What are the two special limit laws?

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1) If c is a constant, then lim┬(x→a) c = c

2 ) lim┬(x→a) x = a By DSP

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## What is Direct Substitution Property? (DSP)

### If f is a polynomial or a rational function and a is in the domain of f, then lim┬(x→a) f(x) = f(a)

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## What is natural numbers denoted by and what does it represent?

### The set of counting numbers (1,2,3,4,5) including 0 and is denoted by N

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## What is integers denoted by and what does it represent?

### The set of natural numbers with their negatives. (-1, 0, 1) and is denoted by Z

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## What is rational numbers denoted by and what does it represent?

### the set of ratios of integers and is denoted by Q

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## What is real numbers and what is it denoted by?

### They are all rational numbers and numbers in between them. denoted by R

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## Definition for Logarithmic Function.

### The inverse function of the exponential function f(x) = a^x is called the logarithmic function with base a

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