For every x, there is only one y.Example: A circle is not a function because it does not pass the vertical line test.* In a circle, every x can yield two y values
B. Polynomial Functions
A polynomial expression can be written in the form:anXn + an-1Xn-1 + ... + a1X + a0where n is a nonnegative integer called the degree of the polynomial.The leading coefficients are typically real numbers, and the leading coefficient can not equal 0.
C. Rational Functions
A rational expression can be expressed in the form: polynomial in x---------------------------------nonzero polynomial in x*Irrational numbers such as the square root of 2 are permissible.*All polynomials are rational expressions because you can place then on top of a denominator of 1.Ex: (x7 + x) can be written as (x7 + x)/1
D. Algebraic Functions
An algebraic expression in x looks like a rational expression, except that radicals and exponents that are noninteger rational numbers ( such as 5/7) are allowed even when x appears in a radicand or in a base (but not in an exponent).Ex: square root of x.*All rational expressions are algebraic.
E. Domain and Range
The domain of a function f, abbreviated Dom(f), is the set of all "legal" inputs.The range of f is then the set of all resulting outputs.Set builder form for range of X2{y E R I y > or equal to 0}Interval form: [0, infinity)
Solving for the domain of a radical function
* Set the radical greater or equal to 0, we can not take even roots of negative numbers. Solve.* When the radical is in the denominator, take into account that we can not have the denominator equal zero, that would lead to undefined.
Types of intervals
(5,7) and (3, infinity) are examples of open intervals, because they exclude their endpoints. (5,7) is a bounded interval, because it is trapped between two numbers.(3, infinity) is an unbounded interval.[5,7] is an example of a closed interval, because it includes its endpoints, and it is bounded.
The union symbol
The union symbol, U, is used to separate intervals in the event that a number or numbers need to be skipped.
Constant functions
y=3, this graph crosses the y-axisx=3, this graph crosses the x-axis
Graphs to know:y = square root of x
y = square root of x
Graphing Techniques:
Ex: f(x) = x2 + 1This graph is shifted up one unit.
I. Functions that are even/odd/ neither; symmetryEven
A function f is even if and only if f(-x) = f(x)In other words, if you plug in a (-x) into a function, the function remains unchanged.Ex: f(x) = -3x2 + 4f(-x) = -3(-x)2 + 4 = -3x2 + 4The graph of y = f(x) is symmetric about the y-axis. More formally, the point (x,y) lies on the graph if and only if the point (-x,y) does.The term "even function" may have come from the following fact:If f(x) = xn, where n is an even integer, then f is an even function.The reciprocal of a nonzero even function is even.Ex: The functions for both x2 and x-2 (which equals 1/x2) are even.Also, cosx and its reciprocal function secx are even because they are symmetric about the y-axis.
Odd
A function f is odd if and only if f(-x) = -f(x)In other words, if you plug in a (-x) into a function, it would be as if you multiplied by -1.Ex: f(x) = 2x3 - 4xf(-x) = 2(-x)3 - 4(-x) = -2x3 + 4xThe graph of y= f(x) is symmetric about the origin.This means that, if the entire graph is rotated 180 degrees about the origin, we obtain it again.*If the graph has a y-intercept, it must be at the origin.More formally, the point (x,y) lies on the graph if and only if the point (-x,-y) does.The term "odd function" may have come from the following fact:
If f(x) = xn, where n is an odd integer, then f is an odd function.The reciprocal of a nonzero odd function is odd.
Ex: The functions for both x1 (which equals x) and x-1 (which equals 1/x) are odd.Also, sinx, its reciprocal function cscx, in addition to tanx and cotx are all odd functions because they are symmetric about the origin.
Neither even nor odd
Ex: f(x) =2x3 - 3x2 - 4x +4f(-x) = 2(-x)3 - 3(-x)2 - 4(-x) + 4 = -2x3 - 3x2 + 4x +4This is neither what I started with nor the exact opposite of what I started with.Also, not that the degrees are both even and odd.Another note:The zero function (over various domains that are symmetric about 0) is the only function that is both even and odd.*The graph looks like y=0
Topic 2: Trigonometry IConverting
Convert 45 degrees into radians45 degrees = 45 x (pi [rad] /180) = (pi/4) [rad]
Coterminal Angles
Standard angles that share the same terminal side are called coterminal angles. They differ by an integer number of full revolutions counterclockwise or clockwise.If the angle is measured in radians, then its coterminal angles are of the form:(theta) + 2(pi)n, where n is any integer.If the angle is measured in degrees, then its coterminal angles are of the form:(theta) + 360ndegrees, where n is any integer.
How to define trig functions
SOH-CAH-TOASin(theta)=Opp/HypCos(theta)=Adj/HypTan(theta)=Opp/AdjWe can define more trig functions by taking the reciprocal of these. We end up with Csc, Sec, and Cot.Also,tan(theta)=sin/cos=rise/run=slope of terminal side.
"Brothers"
Brothers are angles that have the same reference angle.For example, the angles of 30, 150, 210, and 330 degrees are brothers; they all have the same reference angle of 30 degrees, or (Pi/6) radians.Famous Positive Brothers(Pi/6), (Pi/4), and (Pi/3)Again, for (Pi/6)...look at the patternQ1: (Pi/6)Q2: (5Pi/6); observe that 5 is 1 less than 6Q3: (7Pi/6); observe that 7 is 1 more than 6Q4: (11Pi/6); observe that 11 is 1 less than twice 6This pattern follows the others:Q1: (Pi/4)Q2: (3Pi/4)Q3: (5Pi/4)Q4: (7Pi/4)and Q1: (Pi/3)Q2: (2Pi/3)Q3: (4Pi/3)Q4: (5Pi/3)
How do signs of trig values differ between quadrants?
Remember: "All Students Take Calculus"* All of the six basic trig functions are positive in Q1* Sin and its reciprocal, Csc, are positive in Q2* Tan and its reciprocal, Cot, are positive in Q3* Cos and its reciprocal, Sec, are positive in Q4
Topic 3: Trigonometry IIPart A: Fundamental Trig Identities (IDs)Memorize these in both "directions" (i.e., left-to-right and right-to-left)Reciprocal Identities:
sin2x + cos2x = 11 + cot2x = csc2xtan2x + 1 = sec2x*Tip: The 2nd and 3rd IDs can be obtained by dividing both sides of the 1st ID by sin2x and cos2x, respectively.
What are the Cofunction Identities?
If x is measured in radians, then:sinx = cos ((pi/2)-x)cosx = sin ((pi/2)-x)Think: Cofunctions of complementary angles are equal.
Even/Odd (or Negative Angle) Identities
Among the six basic trig functions, cos (and its reciprocal, sec) are even:cos(-x) = cosxsec(-x) = secx, when both sides are definedHowever, the other four (sin and csc, tan and cot) are odd:sin (-x) = -sinx csc (-x) = -cscx, when both sides are definedtan (-x) = -tanx, when both sides are definedcot (-x) = -cotx, when both sides are definedNote: If f is an even function (such as cos), then the graph of y=f(x) is symmetric about the y-axis.Note: If f is an odd function (such as sin), then the graph of y=f(x) is symmetric about the origin.
Part C: Graphs of the six basic trig functionsRefer to index card of graphs and domains/ranges.
***The VAs in the graph of y = cscs are drawn through the x-intercepts of the graph of y=sinx. This is because cscx is undefined if and only if sinx =01/sinx ; sinx can not equal to zero.***The reciprocals of 1 and -1 are themselves, so cscx and sinx take on each of those values simultaneously. This explains how their graphs intersect.
Part E: Advanced Trig Identities (IDs)***Memorize***Group 1: Sum Identities
sin(u + v) = sinu cosv + cosu sinvThink: "Sum of the mixed-up products"cos(u + v) =cosu cosv - sinu sinvThink: "Cosines [product] - Sines [product]"tan(u + v) = tanu + tanv ---------------- 1-tanu tanvThink: " Sum/ (1-Product)"
Group 2: Difference Identities
Simply take the Sum Identities above and change every sign in sight!sin(u - v) = sinu cosv - cosu sinvcos(u - v) =cosu cosv + sinu sinvtan(u - v) = tanu - tanv
----------------
1+tanu tanv
Group 3a: Double-Angle Identities (Think: Angle-Reducing, if u>0)
Also be prepared to recognize and know these "right-to-left."sin(2u) = 2sinu cosuThink: "Twice the product"Reading "right-to-left," we have:2sinucosu = sin(2u)cos(2u) = cos2u - sin2uThink: "Cosines - Sines" (again)Reading "right-to-left," we have:cos2u - sin2u = cos(2u)tan(2u) = 2tanu ----------- 1-tan2uNotice that these identities are "angle-reducing" (if u>0) in that they allow you to go from trig functions of (2u) to trig functions of simply u.
Group 3b: Double-Angle Identities For cosMemorize these three versions of the double-angle identity for cos(2u)
Let's begin with the version we've already seen:Version 1: cos(2u) = cos2u - sin2uAlso know these two, from "left-to-right," and from "right-to-left"Version 2: cos(2u) = 1-2sin2uVersion 3: cos(2u) =2cos2u-1
Group 4: Power-Reducing Identities ("PRIs")
(These are called the "half-angle formulas" in some books"sin2u = 1-cos(2u) 1 1 ---------------- or --- - ---cos(2u) 2 2 2cos2u = 1+cos(2u) 1 1
---------------- or --- + --- cos(2u)
2 2 2tan2u= sin2u 1 - cos(2u) ---------- = ----------------- cos2u 1 + cos(2u)
