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Calculus AB > 12 Basic Functions > Flashcards

12 Basic Functions Flashcards

1
Q

How do you test if a graph is a function

A

Vertical line test

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2
Q

What is a one to one function

A

One or fewer y values for every x value and one or fewer x values for every y value

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3
Q

How do you test if something is a one to one function

A

Horizontal and vertical line tests

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4
Q

What is an asymptote

A

Line which the function comes infinitely close to but never touches

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5
Q

Where are asymptotes commonly found

A

X- and y-axes

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6
Q

What is an odd function

A

A function where the y of -X is equal to the -y of X

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7
Q

What is a characteristic of the graph of an odd function

A

Symmetric about the origin

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8
Q

What is an even function

A

A function where the y of -X is equal to the y of x

This makes the graph symmetric about the y-axis

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9
Q

What does continuous mean

A

The graph has no gaps or stops

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10
Q

What is the relationship between the graphs of inverse functions

A

They are flipped over the line y=x

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11
Q

What is true of the equations of inverse functions

A

If you plug one into the other ‘X’ will come out (i.e. f(g(f))=X AND g(f(X))=X

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12
Q

What is the notation for inverse functions

A

f(X) and f-1(X)

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13
Q

Monotonic concave up/down

Monotonic increasing/decreasing

A

Doesn’t change concavity

Doesn’t change direction

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14
Q

Increasing

A

Going up from left to right

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15
Q

Concave up

A

Bowl shape

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16
Q

Concave down

A

Hat shape

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17
Q

How to find the inverse of a function

A

Y=x2

Flip: X=y2

Solve: y=square root of X

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18
Q

Oscillating

A

Constantly changing concavity (eg sine and cosine functions)

19
Q

Definition of a function

A

One or fewer y values for every x value

20
Q

Which one is domain and which is range?

A

Domain: Possible X values

Range: Possible Y values

21
Q

What causes horizontal asymptotes

A

End behavior

22
Q

Describe the graph and equation of:

The Identity Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes

A

Graph= line f(x)=x

Domain: AR# Range: AR#

Odd

Continuous

One-to-One

Not an inverse

No asymptotes

23
Q

Describe the graph and equation of:

The Squaring Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

Graph=parabola f(x)=x2

Domain: AR# Range:[0, +∞)

Even

Continuous

Not one-to-one

Inverse of Square root function

No asymptotes

24
Q

Describe the graph and equation of:

The Cubing Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes

A

Vertical squiggle: f(x)=x3

Domain: AR# Range:AR#

Odd

Continuous

One-to-One

Not an inverse

No Asymptotes

25
Q

Describe the graph and equation of:

The Reciprocal Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

Diagonal hyperbola: f(x)= 1/x

Domain: x≠0 Range y≠0

Odd

NOT Continuous

One-to-one

Not an inverse

Asymptotes at both axes

26
Q

Describe the graph and equation of:

The Square Root Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

Sideways half-parabola: f(x)=√x

Domain: [0, +∞) Range: [0, +∞)

Neither even nor odd

Continuous

One-to-one

Inverse of squaring function

No asymptotes

27
Q

Describe the graph and equation of:

The Exponential Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

J-shape: f(x)=ex

Domain: AR# Range: (0, +∞)

Neither even nor odd

Continuous

One-to-one

Not an inverse

Asymptote on x-axis (y=0)

28
Q

Describe the graph and equation of:

The Natural Logarithm Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

Curve facing down/left: f(x)=ln(x)

Domain: (0, +∞) Range: AR#

Neither even nor odd

Continuous

One-to-one

Not an inverse

Asymptote @ y-axis (x=0)

29
Q

Describe the graph and equation of:

The Sine Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

Horizontal squiggle: f(x)=sin(x)

Domain: AR# Range: [-1, 1]

Odd

Continuous

Not one to one

Not an inverse

No asymptotes

30
Q

Describe the graph and equation of:

The Cosine Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

Hoziontal squiggle: f(x)= cos(x)

Domain: AR# Range: [-1, 1]

Even

Continuous

Not one-to-one

Not an inverse

No asymptotes

31
Q

Describe the graph and equation of:

The Absolute Value Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

Large V: f(x)=|x| (aka abs(x)

Domain: AR# Range [0, +∞)

Even

Continuous

Not one-to-one

Not an inverse

No asymptotes

nb Min. point= Cusp

32
Q

Describe the graph and equation of:

The Greatest Integer Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

Steps increasing left->right: f(x)=int x

int x=largest whole number ≤ x

Domain: AR# Range: All integers

Neither even nor odd

NOT Continuous

Not one-to-one

Not an inverse

No asymptotes

nb aka Piecewise/step function

33
Q

Describe the graph and equation of:

The Logistic Function

Domain and range?

Even or odd?

Continuous?

One-to-One?

Inverse of another basic function?

Asymptotes?

A

Increase then plateau: f(x)= 1/(1+e-x)

Domain: AR# Range: (0, 1)

Neither even nor odd

Continuous

One-to-one

Not an inverse

Asymptotes @ x-axis (y=0) and y=1

34
Q

Talk about limits and give an example of a graph which has a limit of

f(x)=0 as x→+∞

f(x)=1 as x→+∞

f(x)=0 as x→-∞

A

A limit is another word for an asymptote which gives more information by indicating from which direction the function approaches it.

Becaues the reciprocal function approaches its x-axis asymptote as x increases to the right, it would have a limit of f(x)=0 as x→+∞

Because the logistic function approaches its asymptote at y=1 as x increases from left to right, it would have a limit of f(x)=1 as x→+∞

Because the exponential function approaches its asymptote on the x-axis as x decreases to the left, it would have a limit at f(x)=0 as x→-∞

35
Q

How to find x-and y-intercepts from the equation of a function? What are the intercepts of the natural logatrithm function?

A

Plug 0 in for y to find the x-intercepts and plug 0 into x to find the y intercepts.

The equation for the natural logarithm function is y=ln(x). Due to its vertical asymptote at the y-axis it has no y-intercept, but to find its x-intercept I will plug 0 in for y

0=ln(x)

e0=x

1=x

36
Q

Options for solving quadratics w/instructions

A

1: No matter what, immediately get one side to equal 0
2: If possible, factor

x2+5x+6=0

(x+3)(x+2)=0

x=-3 or x=-2

2: If factoring’s too hard, Quadratic formula

-5±√(5)2-(4✖1✖6)

2(1)

-5±√1

2

-5+√1 = -2=x or -5-√1 = -3=x

2 2

  1. If all else fails, complete the square

x2 – 4x – 8 = 0

x2 – 4x=8

x2 – 4x + 4= 8+4=12

(x-2)2=12

x-2= +√12

x=2+√12 or x=2-√12

x=2+2√3 x=2-2√3

37
Q

Rules never to break when solving quadratics

A

DON’T divide by X

DON’T forget ± when √

38
Q

Volume of:

Cone

Pyramid

Sphere

A

Cone: 1/3πr2h

Pyramid: 1/3(area of base)h

Sphere:4/3πr3

39
Q

Surface area of a sphere

A

4πr2

40
Q

Circumference and area of a circle

A

Circumference: πd

Area: πr2

41
Q

Generic interest/growth formula

A

A=P(1+(r/n))nt

r=decimal of %rate (eg 10%=.10)

t=number of years

n=times compounded per year

Continuous:

A=Pert

e=Mathematical constant

r=decimal rate

t=number of years

42
Q

How to do half life

A

P(.5)t/half life=amount

Example: 6.6g initial 14 day half life

How long til 1 gram?

6.6(.5)t/14=1

ln(6.6) + ln(.5)(t/14)=ln(1)

ln(6.6) + ln(.5)(t/14)=0

ln(.5)(t/14)=-ln(6.6)

t/14=(-ln(6.6))/(ln(.5)) x 14

t=38.11 days!

43
Q

What is an inflection point

A

The point where the concavity of a graph changes

44
Q

Which kinds of parentheses are inclusive and not inclusive? How woud you represent the domain and range of the logistic function?

A

[…….] = inclusive

(……) = not inclusive

The domain of the logistic function is all real numbers, which could be stated either with words or by writing

(-∞, +∞)

nb Infinity is always not inclusive

The range of the logistic function is between y=0 and y=1, not inclusive, which would be written

(0, 1)

Calculus AB (3 decks)

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