Study Flashcards
Even functions are symmetric with respect to
y-axis
Symmetric with respect to y-axis
Even function
Odd functions are symmetric with respect to
origin
Symmetric with respect to the origin
Odd function
A function is not a polynomial when
Power is not an integer
When a power is not an integer in a polynomial
It is not a polynomial
What is the domain of tanx?
{x € R | x ≠ π/2 + nπ}
{x € R | x ≠ π/2 + nπ}
Domain of tanx
How do you prove a function is odd?
Place (-x) in for x
Place (-x) in place of x to
Prove a function is odd
Constants are what type of function
Even
Can even functions be constants?
Yes
Functions that are not even or odd are not symmetric across
y-axis OR origin
Not symmetric across y-axis OR origin
Functions that are neither even nor odd
Rules of graphing inverse of a function
Reflect original over line y=x and any point (a,b) on f(x) equals
(b, a) on f^-1(x)
Reflect original over line y=x and any point (a,b) on f(x) equals
(b, a) on f^-1(x)
Rules of graphing inverse of a function
y = log(a)x equals
a^y = x
a^y = x equals
log(a)x
Every log function hits the point what on graph
(1, 0)
Always hits point (1,0) on graph
Log functions
What is the second point on the graph of log(5)x and why?
(5,1) because the base is five, and 5^1 equals 5. So the second five is the x and y is what it is raised to
log(a)AB equals
log(a)A + log(a)B
log(a)A + log(a)B equals
log(a)AB
log(a)A/B equals
log(a)A - log(a)B
log(a)A - log(a)B equals
log(a)A/B
log(a)A^c equals
c log(a)A
c log(a)A
log(a)A^c
Change of base formula
log(b)x = log(a)x/log(a)b
log(b)x = log(a)x/log(a)b
Change of base formula
To graph a function like this:
y=-2(x-3)^1 (x+1)^3 (x+4)^1
What is done?
Add of exponents and consider negative sign in front. It will be similar to the graph of -x^5 but with curved points
How do you graph this function?:
y=3-2 √4-x
Plug in three numbers for x that make the radical square-root-able and gets u values
For a composite function (f o g), the domain is
Defined when both f(x) and f(g(x)) are defined.
Defined when both f(x) and f(g(x)) are defined.
For a composite function (f o g)
How would you prove (f o g)(x) = (g o f)(x)?
Find domains of each and compare.
What does [cos(x+9)]^2 equal?
cos^2(x+9)
What does cos^2(x+9) equal?
[cos(x+9)]^2
What is the graph of 1/x^2 like?
Two hyperbolas one in quadrant 1 and one in quadrant 2
Two hyperbolas one in quadrant 1 and one in quadrant 2
Graph of 1/x^2
Find vertical asymptotes of a function
Set denominator equal to zero
Set denominator equal to zero
Find vertical asymptotes of a function
How to find x values of:
y=log(3)(x-1) + 2
Use inside of parentheses:
x-1 = 0 (vertical asymptote) x-1 = 1 (first x value) x-1 = 3 (base)
Plug these in to original function to find y values
How do you graph:
y=(1/3)^x
Know that this equals y=3^-x:
- x=0=0 (y-intercept)
- x=-=-1 (second point)
So, points at (0,1) and (-1,3)
b^(x+y) equals
b^x • b^y
b^x • b^y equals
b^(x+y)
b^(x-y) equals
b^x / b^y
b^x / b^y equals
b^(x-y)
(ab)^x equals
a^x • b^x
a^x • b^x equals
(ab)^x
Horizontal line test can test what type of function?
One-to-one
How to tell if the graph if a function is one-to-one?
Horizontal line test
How is an inverse function defined?
f(x) = y <=> f^-1(y) = x
f(x) = y <=> f^-1(y) = x
Definition of inverse function
What do f(f^-1(x)) and f^-1(f(x)) equal?
x
How is log(2)(16) evaluated?
Have to find number common with base so:
log(2)(2^4) = 4log(2)(2) = 4
What does y=ln(x) equal?
e^y=x
What does e^y=x equal?
y=ln(x)
Change of base formula (natural logs)
log(b)(x)=ln(x) / ln(b)
log(b)(x)=ln(x) / ln(b)
Change of base formula (natural logs)
How do you evaluate log(8)(5)?
Change of base formula:
log(8)(5)= ln(5) / ln(8)
If a rectangle has a perimeter of 24 ft and you need to find a function that models its area A in terms of the length L of one of its sides, what do you do?
Make L the only variable:
P=2L+2w and since
P=24, that means 24=2L+2w and w=12-L.
Since A=Lw, this means A(L)=L(12-L).
Make L the only variable:
P=2L+2w and since
P=24, that means 24=2L+2w and w=12-L.
Since A=Lw, this means A(L)=L(12-L).
Expressing area of a rectangle in terms of the length of one of its sides.
What is the domain of the area of a rectangle as a function of length when assuming width < length?
Let A(L)=L(12-L). The domain would be (6,12) because if L=6, width would be 6. If L=12, width would be 0.
Let A(L)=L(12-L). The domain would be (6,12) because if L=6, width would be 6. If L=12, width would be 0.
Domain of the area of a rectangle as a function of length when assuming width < length.
Area and domain of an equilateral triangle.
A(x)= √3/4(x^2) where the domain is (0, ∞) because a side cannot be negative.
A(x)= √3/4(x^2) where the domain is (0, ∞) because a side cannot be negative.
Area and domain of an equilateral triangle
What happens when you get the number √2/7 where the whole thing is square rooted?
It also means √2/√7 so rationalize by multiplying by √7 to get √14(over)7
Order of transformations
H-shift H-stretch/shrink Reflect y-axis Vert-stretch/shrink Reflect x-axis Vert-shift
H-shift H-stretch/shrink Reflect y-axis Vert-stretch/shrink Reflect x-axis Vert-shift
Order of transformations
How do you reflect the graph of y=e^x over the line y=7 and x=3?
For y-values, add negative to coefficient and add double the y-value. For x-values, add negative to x and add double the x-value to x.
For y-values, add negative to coefficient and add double the y-value. For x-values, add negative to x and add double the x-value to x.
Reflecting graphs over y lines and x lines
