12 Basic Functions Flashcards
How do you test if a graph is a function
Vertical line test
What is a one to one function
One or fewer y values for every x value and one or fewer x values for every y value
How do you test if something is a one to one function
Horizontal and vertical line tests
What is an asymptote
Line which the function comes infinitely close to but never touches
Where are asymptotes commonly found
X- and y-axes
What is an odd function
A function where the y of -X is equal to the -y of X
What is a characteristic of the graph of an odd function
Symmetric about the origin
What is an even function
A function where the y of -X is equal to the y of x
This makes the graph symmetric about the y-axis
What does continuous mean
The graph has no gaps or stops
What is the relationship between the graphs of inverse functions
They are flipped over the line y=x
What is true of the equations of inverse functions
If you plug one into the other ‘X’ will come out (i.e. f(g(f))=X AND g(f(X))=X
What is the notation for inverse functions
f(X) and f-1(X)
Monotonic concave up/down
Monotonic increasing/decreasing
Doesn’t change concavity
Doesn’t change direction
Increasing
Going up from left to right
Concave up
Bowl shape
Concave down
Hat shape
How to find the inverse of a function
Y=x2
Flip: X=y2
Solve: y=square root of X
Oscillating
Constantly changing concavity (eg sine and cosine functions)
Definition of a function
One or fewer y values for every x value
Which one is domain and which is range?
Domain: Possible X values
Range: Possible Y values
What causes horizontal asymptotes
End behavior
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
Graph= line f(x)=x
Domain: AR# Range: AR#
Odd
Continuous
One-to-One
Not an inverse
No asymptotes
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?
Graph=parabola f(x)=x2
Domain: AR# Range:[0, +∞)
Even
Continuous
Not one-to-one
Inverse of Square root function
No asymptotes
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
Vertical squiggle: f(x)=x3
Domain: AR# Range:AR#
Odd
Continuous
One-to-One
Not an inverse
No Asymptotes
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?
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
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?
Sideways half-parabola: f(x)=√x
Domain: [0, +∞) Range: [0, +∞)
Neither even nor odd
Continuous
One-to-one
Inverse of squaring function
No asymptotes
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?
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)
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?
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)
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?
Horizontal squiggle: f(x)=sin(x)
Domain: AR# Range: [-1, 1]
Odd
Continuous
Not one to one
Not an inverse
No asymptotes
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?
Hoziontal squiggle: f(x)= cos(x)
Domain: AR# Range: [-1, 1]
Even
Continuous
Not one-to-one
Not an inverse
No asymptotes
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?
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
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?
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
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?
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
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 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→-∞
How to find x-and y-intercepts from the equation of a function? What are the intercepts of the natural logatrithm function?
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
Options for solving quadratics w/instructions
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
- 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
Rules never to break when solving quadratics
DON’T divide by X
DON’T forget ± when √
Volume of:
Cone
Pyramid
Sphere
Cone: 1/3πr2h
Pyramid: 1/3(area of base)h
Sphere:4/3πr3
Surface area of a sphere
4πr2
Circumference and area of a circle
Circumference: πd
Area: πr2
Generic interest/growth formula
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
How to do half life
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!
What is an inflection point
The point where the concavity of a graph changes
Which kinds of parentheses are inclusive and not inclusive? How woud you represent the domain and range of the logistic function?
[…….] = 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)
