Functions- lecture video Flashcards
How many outcomes can a function have
one
each input determines one unique output
What is the Domain
x value or input value or independent variable
what is the range
y value or output value or dependent variable
What test can we use to determine if a graph is a function
vertical line test
- the vertical line can only pass through one co-ordinate
is this a function? Why or why not
X^2 + y^2 = 25
no, we need to
at 15:15 add more
but you end up with more than one answer +/_ so it is not a function
What is important about graphing piecewise functions
by graphing each piece individually
What are the restrictions for A=S^2
S must be greater than or = to zero
(cannot be a negative number)
what are the restrictions for y = 1/ x
x cannot = 0
0 would make it undefined
what are the restrictions for F(x) = the square root of x
x must be greater than or = to zero
what are the restrictions for f(x) = x ^3
no restrictions, so the domain is all real numbers
what are the restrictions for f(x) = 1 / (x-1) (x-3)
X is all really numbers except x cannot = 1 or 3
What happens if you cannot simplify restrictions out of your functions
these will be vertical asymptotes
What are the restrictions for f(x) = Tan x
(an s shape for tangents, because it is undefined at certain points)
= sin x / cos x
therefore, cos x cannot = 0
x cannot = pi/ 2
x cannot = 3 pi / 2
etc
what are the restrictions for f(x) = square root (x^2-5x +6)
- if you have a square root, the numbers must be positive in it
- we know that (x^2 - 5x + 6 must be greater than or = to zero)
- We need to factor this question because it is a quadratic equation
= (x - 3)(x-2) is greater than or = 0
this is an inequality, becareful, it’s not 3 and 2
- so we know x = 2 and x = 3 are important points, make a number line and put the points on the number line then do a test
- here we plug in 0 (to the left of 2) into the equation, you get positive 6
- every number to the left of 2, will be a positive number - try a point greater than 3, we plug in 4 and we get a positive number
- try a point b/w 2 and 3, we choose 2.5 and we get a negative number
Therfore the domain is:
(- infinity,2]U [3, + infinity)
Big thing with restrictions is
look for
1. denominators
- the bottom cannot = 0
- roots
- cannot use negative numbers as the answer
What are the restrictions for f (x) = x^2 - 4 / x-2
- we know the denominator cannot be 0 however, we can factor this
- factor this out we get
(x+2) (x-2) /x-2
- we can now cancel out the two x-2s and we get
x+2 - so the domain is all real numbers except we still have to keep the original domain restriction that x cannot = 2
If the restriction is x cannot = 2 what also can we say about x = 2 on the graph
we have a point on the graph not filled in or a hole
or a removable discontinuity (or not continuous graph)
If you can cancel out your domain problem (ie. the denominator can be cancelled out by factoring) then you have what
a hole
If you cannot cancel out your domain problem (ie. the denominator by factoring) what is it
an asymptote
What are the restrictions for 3x / x-4
do you have a hole or asymptote
- x cannot = 4
- we cannot factor the denominator out so we have a vertical asymptote
what are the restrictions for
f(x) = 2 + square root of x-1
What is the domain and what is the range
x-1 cannot = be negative or must be greater than or = to zero
so x must be greater than or equal to 1
Domain:
interval
[1, infinity)
Range: [2, infinity)
how do you find your range?
plug in your domain (for now, for simple items)
what is the domain and range for y = (x+1)/ (x-1)
Domain: x cannot = 1
bc 1 -1 =0 in the denominator
all real numbers except 1
now plug in 1 into the whole equation you get
2/ 0 which is a vertical asymptote
Range: if you solve for y you get
x = y+1 / y-1
y cannot = 1
horizontal asymptote
What are even functions
functions that have 2s, 4s, 6s, and 8,s in them etc
What are odd functions
functions that have odd numbers in them
even functions are symmetric how
symmetric across the y-axis
f(-x) = f(x)
odd functions are symmetric how
symmetric about the origin (mirror image)
f(-x) = -f(x)
how to test to see if something is even
If f(x) = x^4 - x^2+1
plug in negative x and see what happens
f(-x) = (-x)^4 - (-x)^2+1
becomes…
x^4-x^2+1 - we got back the same thing!
THis means it is an EVEN function
is it even or odd function
f(x) = x^3 - x
to figure out, plug in -x
f(-x) = (-x)^3 - (-X)
becomes….
F(-x)= -x^3 +1
We got the opposite back so this means it is
an ODD Function