Unit 2 Overview

When you have finished this unit, you will be able to do the following:

Find the domain and range of a relation

Determine whether a relation is a function

Determine whether an equation represents a function

Evaluate a function

Graph functions by plotting points

Use the vertical line test to identify functions

Obtain information about a function from its graph

Identify the domain and range of a function from its graph

Identify intercepts from a function's graph

Identify intervals on which a function increases, decreases, or is constant

Use graphs to locate relative maxima or minima

Test for symmetry

Identify even or odd functions and recognize their symmetries

Understand and use piecewise functions

Find and simplify a functions difference quotient

Calculate a lines slope

Write the equation of a line in the form of

• point-slope
• slope intercept
• general form

Graph horizontal or vertical lines

Use intercepts to graph a line from general form

Model data with linear functions and make predictions

Find slopes and equations of parallel and perpendicular lines

Interpret slope as rate of change

Find a functions average rate of change

Find the domain of a function

Combine functions using the algebra of functions, specifying domains

Form composite functions

Determine domains for composite functions

Write functions as compositions

Verify inverse functions

Find the inverse of a function

Use the horizontal line test to determine if a function has an inverse function

Use the graph of a one to one function to graph its inverse function

Find the inverse of a function and graph both functions on the same axes

Find the distance between two points

Find the midpoint of a line segment

Write the standard form of a circle's equation

Give the center and radius of a circle whose equation is in standard form

Convert the general form of a circles equation to standard form

The follow section will give a brief overview of each section. It will only talk about the main highlights of the section. Examples may or may not be provided. If you are wanting to see examples, refer to the notes that were done in class.

The mathematical term for a set of ordered pairs is a relation. The set of all first components of the ordered pair is called the domain of the relation and the set of all second components is called the range of the relation.

A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range. A functions is a relation in which no two ordered pairs have the same first component and different second components.

A function can have two different first components with the same second component. By contrast, a relation is not a function when two different ordered pairs have the same first component and different second components.

If components form a function, then there is one and only one value of y. The variable x is called the independent variable because it can be assigned any value from the domain. The variable y is called the dependent variable because its value depends on x.

Nor all equations with the variables x and y define functions. If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a function of x.

Up until this point, all the equations have been in the form of y =. If an equation in x and y gives one and only one value of y for each value of x, then the variable y is a function of the variable x. Variable x is the input or domain, and variable y is the output or range. Think of the domain as the set of the function's inputs and the range as the set of the functions outputs. Because of this mindset, we can then say y = f(x) which reads as f of x or f at x. This shows the output of the value of the function at the number x.

The graph of a function is the graph of its ordered pairs. You will graph a function in the rectangular coordinate plane.

Not every graph in the rectangular coordinate system is the graph of a function. The definition of a function specifies that no value of x can be paired with two or more different values of y. If a graph contains two or more different points with the same first coordinate, the graph cannot represent a function.

If any vertical line intersects a graph in more than one point, the graph does not define y as a function of f.

You can obtain information about a function from its graph. At the right or left of a graph, you will find closed dots, open dots, or arrows.

• A closed dot indicates that the graph does not extend beyond this point and the point belongs to the graph.
• An open dot indicates that the graph does not extend beyond this point and that the point does not belong to the graph.
• An arrow indicates that the graph extends indefinitely in the direction in which the arrow points.

The zeros of a function are where the graph crosses the x-axis. These zeros are also known as x-intercepts. Where a graph crosses the y-axis is called the y-intercept. This happens when the x value equals zero. A function can have more than one x-intercept, but at most, only one y-intercept.

A function is increasing on an open interval as long as y is increasing when x is increasing.

A function is decreasing on an open interval as long as y is decreasing when x is increasing.

A function is constant on an open interval as long as y is the same at any x value on that interval.

The points at which a function changes its increasing or decreasing behavior can be used to find any relative maximum or relative minimum values of the function.

A function value f(a) is a relative maximum of f(x) if the behavior at x = a goes from an increasing behavior to a decreasing behavior.

A function value f(a) is a relative minimum of f(x) if the behavior at x = a goes from a decreasing behavior to an increasing behavior.

The word local is sometimes used instead of relative when describing maxima or minima.

A graph is symmetric to the y - axis if every point (x, y) on the graph also has the point (-x, y). The way you test for this symmetry is when you substitute -x for x in the equation, the y value is the same value for both x and -x.

A graph is symmetric to the x - axis if every point (x, y) on the graph has the point (x, -y). The way that you test for this symmetry is when you substitute -y for y in the equation and solve for x. You should get the same x value back as you did when you solved the same equation for x when you plugged in y.

A graph is symmetric to the origin if every point (x, y) on the graph has the point (-x, -y). The way that you test for this symmetry is when you substitute -x in for x, you get a -y value.

A function whos graph is symmetric with respect to the y-axis is called an even function. If the point (x, y) is on the graph of the function, then the point (-x, y) is also on the graph. Expressing the symmetry in function notation provides the definition of an even function. The right side of the equation of an even function does not change if x is replaced with -x.

A function whose graph is symmetric with respect to the origin is called an odd function. If the point (x, y) is on the graph of the function, then the point (-x, 0y) is also on the graph. Expressing this symmetry in function notation provides the definition of an odd function. Every term on the right side of the equation of an odd function changes sign if x is replaced with -x.

If a function is neither even nor odd, then the right side of the equation changes if x is replaced with -x, but not every term on the right side changes sign.

The difference quotient is a formula you will need to learn if you are going to take Calculus. This will help you find the average rate of change formula for any specific value given f(x). There is a 4 step process to be able to find the equation.

1. Find f(x + h)

2. Simplify f(x + h) - f(x)

3. Take the answer from number 2 and put over h

4. Simplify the function.

The slope, m, of 2 points is the change in y over the change in x.

If you are given a point and the slope or if you are given two different points, you can use the point-slope form to find the equation of a line. This is a valid equation for a line. If you are wanting to graph this line, you would solve for y and put it in slope-intercept form.

Typically, when you graph a line, you graph the line from the form of Slope-Intercept. This is usually in the form of y = mx + b, where m is the slope and b is the y-intercept.

When you use this form to graph a line, follow the following steps:

1. Plot the point containing the y-intercept on the y-axis. This is the point (0, b).
2. Obtain the second point by using the slope from the formula. If the slope is a whole number, that is not in fraction form, the number you see is the rise and you will run 1 to the right. If the slope is already in a fraction form, then the rise is the top number and the run is the bottom number. Place a dot when you get to the new point.
3. Use a straightedge (because lines are not crooked) to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.

When lines are in this format, you will graph this line by using intercepts only. This means you will make x = 0 and solve for y to find the y-intercept and then you will make y = 0 and solve for x to find the x-intercept. Then, you will connect the two points to create the line.

There are some rules when you are in this form.

1. Ax must always be positive. The By can be either positive or negative.
2. There cannot be any fractions in this form. If there are fractions, you must remove the fractions.

A horizontal line is given my an equation in the form of:

y = b

Where b is the y-intercept of the line. The slope of a horizontal line is zero. This linear function is often called a constant function.

A vertical line is given by an equation in the form on:

x = a

Where a is the x-intercept of the line. The slope of a vertical line is undefined.

No vertical line represents a linear function!

Two nonintersecting lines that lie in the same plane are parallel. If two lines do not intersect, the ratio of the vertical change to the horizontal change is the same for both lines. Because two parallel lines have the same steepness, they must have the same slope.

Two lines that intersect at a right angle are said to be perpendicular. The relationship between the slopes of perpendicular lines is not as obvious as the relationship between parallel lines. The relationship between the two slope are that they are negative reciprocals of each other.

1. If two nonvertical lines are parallel, then they have the same slope.
2. If two distinct nonvertical lines have the same slope, then they are parallel.
3. Two distinct vertical lines, both with undefined slopes, are parallel.

1. If two nonvertical lines are perpendicular, then the product of their slopes is -1.
2. If the product of the slopes of two lines is -1, then the lines are perpendicular.
3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.

If the graph of a function is not a straight line, the average rate of change between any two points is the slope of the line containing the two points. This line that will cut through the function is called the secant line.

Functions that model data often have their domains explicitly given with the function's equation. However, for most functions, only an equation is given and the domain is not specified. In cases like this, the domain of a function is the largest set of the real numbers for which the value of f(x) is a real number.

If a function, f, does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function's domain real numbers that cause division by zero and real numbers that result in an even root, such as a square root, of a negative number.

We can combine functions using addition, subtraction, multiplication, and division by performing operations with the algebraic expressions that appear on the right side of the equations. Given f(x) and g(x), the following operations can happen.

The domains for each of these will be a combination of the domains of f(x) and g(x).

Composing functions means to take one function and place it into each variable of the other function. If you are given f(x) and g(x), then you can compose these two functions and create a new function.

The domain of a composite function is the set of all x such that

1. x is in the domain of g
2. g(x) is in the domain of f

or

1. x is in the domain of f
2. f(x) is in the domain of g

This means that you need to be careful in determining the domain for a composite function. If you are looking at the composite function f(g(x)), then the following values must be excluded from the input x:

1. If x is not in the domain of g, it must not be in the domain of f(g(x))
2. Any x for which g(x) is not in the domain of f must not be in the domain of f(g(x)).

NOTE: f(g(x)) is NOT the same thing as g(f(x)).

If two functions are true inverses of each other, then the following must hold true:

• f(g(x)) = x
• g(f(x)) = x

1. If there is an f(x), replace f(x) with y.
2. Interchange x and y.
3. Solve for y
4. Put the appropriate symbol for inverse in place of the y.

A function f has an inverse a function if there is no horizontal line that intersects the graph of the function f at more than one point. If the line only crosses once, then it passes the horizontal line test and that function has an inverse. A one-to-one function is a function in which no two different ordered pairs have the same second component. Any function that passes the horizontal line test is a one-to-one function and any one-to-one function has a graph that has passed the horizontal line test. If this is true, then the original graph and its inverse will be reflected across the y = x line.

To compute the distance between two points, find the square of the difference between the x-coordinates plus the square of the difference between the y-coordinates. The principal square root of this sum is the distance.

If you are given a line segment, take the average of the two x-coordinates and the average of the two y-coordinates. This will be the midpoint of that line segment.

A circle is the set of all points in a plane that are equidistant from a fixed point, called the center. The fixed distance from the circle's center to any point on the circle is called the radius.

When you are given the center (h, k) and the radius (r), you can use this information and put it into the standard equation of a circle.

The general form of the equation of a circle is

where D, E, and F are real numbers.

The convert General Form to Standard Form, you must complete the square....twice!