PreCal

Chapter 1.1

Relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in the set, called the range.

Function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactly one element in the second set, called the range.

Example:

Function

3- 3 4-

y= x2 is an example of a function and on the graph looks as parabola

Relation

--4 3 ---4

y= y2 is an example of relation

Vertical line test! Given the graph of an equation, if any vertical line that can be drawn intersects the graph at no more than one point, the equation defines a function of x.

Functions that are used in application often have restrictions on the domains due to physical constrains.

f(x)= square root of x is an example of implicit domain.

Chapter 1.2

The nine main functions:

1)Linear function. f(x) = mx + b. Where m and b are real numbers

2)constant function f(x) = b. Where b is any real number It is a horizontal line. The y int corresponds to e point (o, b)

3) identity function. F(x) = x It passes through the origin and every point that lies on the line has equal x and y coordinates. It is the set of all real numbers

4) square function f(x)= x2

The graph of this function is a parabola. It is the set of all real numbers.

5) cube function f(x) = x3

The set of all real numbers because curbing a negative number yields a negative number, cubing a positive number yields a positive number, and cubing 0 yields 0, the range of the cube function is also the set of all real numbers.

6) square root function f(x) = square root of x

The. Output of e function will be all real numbers greater than or equal to zero.

7) cube root function f(x) = square root of x 3

This graph would contain in quadrants I and III and passes through the origin. Tis function is symmetric about the origin.

8) absolute value function. f(x) = IxI

It's the set of all real numbers, yet the range is the set of nonnegative real numbers.

9) reciprocal function f(x) = 1/x. x can not equals zero

The set of all real numbers excluding zero.

Chapter 1.3

Vertical shifts:

Assuming that c is a positive constant,

To graph Shift the Graph of f(x) f(x)+ c c units upward f(x)- c c units downword

Adding or subtracting a constant outside the function corresponds to a vertical shift that goes with the sign

Horizontal shifts:

Assuming that c is a positive constant,

To Grapg f(x+c) c units to the left f(x-c) c units to the right

Adding or subtracting a constant inside the function corresponds to a horizontal shift that goes opposite the sign

Reflection about the Axes The graph of -f(x) is obtained by reflecting the graph of f(x) about the x-axis The graph of f(-x) is obtained by reflecting the graph of f(x) about the y-axis

Vertical stretching and Vertical compressing of graphs

The graph of cf(x) is found by: (с is any positive number).
 * Vertically stretching the graph of f(x)c if c > 1
 * Vertically compressing the graph of f(x) if 0<с<1

Horizontal stretching and horizontal compressing of graphs

the graph of f(cx) is found by:

(с is any positive number)
 * Horizontally stretching the graph of f(x) if 0 <с<1
 * Horizontally compressing the graph of f(x) if c > 1

Chapter 1.4

Function Notation Sum (f+g)(x)=f(x) + g(x) Difference (f-g)(x)+f(x) - g(x) Product (f * g)(x)=f(x) * g(x) Quotient (f/g)(x)= f(x)/ g(x) g(x) cant = 0

The domain of the sum, difference, and product functions is the intersection of the domain, or common domain shared by both f and g. THe domain of the quotient function is also the intersection of the domain shared by both f and g with an additional restriction that g(x) cant = 0

Composition of Function

(f*g)(x)=f(g(x)) The domain restrictions cannot always be determined simply by inspecting the final form of f(g(x)). Rather, the domain of the composite function is a subset of the domain of g(x). Values of x must be eliminated if their corresponding values of g(x) are not in the domain of f.

Chapter 1.5

One -to-one Function

A function f(x) is one-to-one if no two elements in the domain correspond to the same element in the reange. That is, if x1 cannot = x2, then f(x1) cannot = f(x2)

Horizontal line test

If every horizontal line intersects the graph of a function in at most one point, then the function is classified as one-to-one function.

Inverse function

If f and g denote two one-t-one functions such that

f((x)) = x for every x in the domain of g and g(f(x)) = x for every x in the domain of f, then g is the inverse of the function f. The function g is is denoted by f-1

Domain of f = range of f-1 Domain of f-1 = range of f

Halloween Word problem;

Americans are expected to shell out almost $6 billion this halloween and more than a third of that will be spent on costumes. This year is expected to be a big year for Halloween. On average, Americans will spend about $25 on their costume. An estimated 2 out of 5 Americans are expected to wear costumes this year. Thats up a third from last year.

1) According to the information above, an estimated two out five Americans are expected to wear a costume this year. if there are about 300000000 people living in the U.S. about how many of them will be wearing a costume this Halloween?

solution:

300000000 multiply by 2/5 = 120000000

2) according to the information above two out of every five Americans will dress up for Halloween. The reading also tells us that this up one third from last year. Use this information to write the ratio of Americans wearing a halloween costume to total Americans for the year before this year?

Solution

2/5 divide by 4/3 = 3/10 __**Chapter 2.1**__ **Polynomial Function -** Let n be a nonnegative integer, and let an, an-1,...., a2, a1, a0 be real numbers with an not equals 0. The function f(x) - anxn + an-1 = >>> _ a2xsquared + a1x + a0 is called a polynomial function of x with degree n. The coefficient an is called the leading coefficient, and a0 is the constant. **Quadratic Function -** Let a, b, and c be real numbers with a not equals 0. The function f(x) = axsquared + bx + c is called a quadratic function. The quadratic Function f(x) = a(x-h)2 + k is in standart form. The graph of of is a parabola whose vertex is the point (h, k). The parabola is symmetric with respect to the line x = h. If a > 0, the parabola opens up. if a < 0, the parabola opens down. When graphing quadratic functions (parabolas), have at least 3 points labeled on the graph.
 * When there are x - intercepts, label the vertex, y - intercept, and x-intercepts
 * When there are no x-intercepts, label the vertex, y-intercept, and another point.
 * Vertex Of Parabola:**

The graph of a quadratic function f(x) = axsquared + bx+ c is a parabola with the verbs located at the point. (-b/2a, f ( -b/2a))


 * Graphing a quadratic function in general form:**

Step 1: Find the vertex Step 2: Determine whether the parabola opens up or down. Step 3:Find additional points near the vertex Step 4: Sketch the graph with a parabolic curve
 * if a >0, the parabola opens up
 * if a<0, the parabola opens down

Mini Project
 * Population Growth: **

N=N0eRT

Where:
 * N0 ** is the starting population
 * N ** is the population after a certain time, **T**, has elapsed
 * R ** is the rate of natural increase expressed as a percentage and
 * E ** is the constant 2.71828... (The base of natural logarithms).

Population grows exponentially - if the rate of natural increase (**r**) doesn't change. The variable **r** is controlled by human behavior.


 * Exponential Calculations: **

Before electronic calculators, logs were used all the time for doing exponential calculations. So scientists and engineers of all kinds made use of logs frequently. For example, if you wanted to find 4^(3.5), you'd use the fact that:

4^(3.5) = 10^Log[4^3.5] = 10^(3.5 * log(4))

You look up log(4) in your log tables, multiply that by 3.5, then use the log table to find the antilog (10 raised to the power of your answer). These days, we usually let calculators do the work, but even the calculators make use of facts like these in order to do the computing.


 * Psychology ** :

Psychological studies found that mathematically unsophisticated individuals tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 20 as 100 is to 200. Increasing mathematical understanding shifts this to a linear estimate (positioning 100 10x as far away)


 * Number Theory: **

Natural logarithms are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theory. For any integer //x//, the quantity of prime numbers less than or equal to //x// is denoted π(//x//). The prime number theorem asserts that π(//x//) is approximately given by x/In(x)


 * Decibels: **

The decibel ( **dB**) is used to measure sound level, but it is also widely used in electronics, signals and communication. The difference in decibels between the two is defined to be: 1- log(P2/P1) dB where the log is to base 10

(Surface Wave Magnitude)
 * Earthquakes: **

IASPEI formula No depth corrections are applied, and Ms magnitudes are not generally computed for depths greater than 50 kilometers. The Ms value published is the average of the individual station magnitudes from reported T and A data. If the uncertainty of the computed depth is considered great enough that the depth could be less than 50 kilometers, an Ms value may still be published, computed by the IASPEI formula and NOT corrected for depth. In general, the Ms magnitude is more reliable than the mb magnitude as a means of yielding the relative "size" of a shallow-focus earthquake.
 * Ms = log (A/T) + 1.66 log D + 3.3 **
 * A ** is the maximum ground amplitude in micrometers (microns) of the vertical component of the surface wave within the period range 18 <= T <= 22.
 * T ** is the period in seconds.
 * D ** is the distance in geocentric degrees (station to epicenter) and 20° <= D <= 160°.