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Chapter 1 – Linear Relations and Functions. 1.1 Relations and Functions Definitions:. A relation is a set of ordered pairs. The domain is the set of all abscissas, x-coordinates, of the ordered pairs. The range is the set of all ordinates, y-coordinates, of the ordered pairs.

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1 1 relations and functions definitions
1.1 Relations and FunctionsDefinitions:
  • A relation is a set of ordered pairs.
  • The domain is the set of all abscissas, x-coordinates, of the ordered pairs.
  • The range is the set of all ordinates, y-coordinates, of the ordered pairs.
  • A function is a relation in which each element of the domain is paired with exactly one element in the range.
slide3
Example: If x is a positive integer less that 6, state relation representing the equation y = 5 + x by listing all ordered pairs. Also state the domain and range.

Vertical line test:

slide4

A function is usually denoted by f. In function notation, the symbol f(x) is interpreted as the value of the function f at x.

Find f(3) if

Find f(4) if

What is the greatest integer function?

2 1 compositions and inverses of functions
2.1 Compositions and Inverses of Functions

Operations with functions:

Sum

Difference

Product

Quotient

slide7

Given f(x) = x + 3 and find the values of each function. Also name all values of x not in the domain.

slide8

Def: Given functions f and g, the composite function can be described by the following equation.

The domain of includes all of the elements x in the domain of g for which g(x) is in the domain of f.

slide9

If and , find

Compose onto itself, this is called an iteration.

slide10

Two functions f and g are inverse functions if and only if

Suppose f and f-1 are inverse functions. Then, f(x) = y if and only if f-1(y) = x

Example: Given f(x) = 4x – 9, find f-1(x) and show that f and f-1 are inverse functions.

slide11

Ex: Given , find f-1(x).

Ex: Given , find f-1(x).

1 3 linear functions and inequalities
1.3 Linear Functions and Inequalities

Solve the following equation:

Def: A linear equation has the form Ax + By + C = 0, where A and B are not both zero. The graph is always a straight line.

Def: Value of x for which f(x) = 0 are called zeros of the function.

1 4 distance and slope
1.4 Distance and Slope

We will now derive the distance formula.

slide17

The slope, m, of the line through (x1,y1) and (x2,y2) is given by the following equation:

Example: Find the distance between (4, -5) and (-2, 3).

Example: Find the slope through (4, 5) and (4, -3).

slide18

If the coordinates of P1 and P2 are (x1,y1) and (x2,y2), respectively,

then the midpoint of the line has coordinates:

Example: Find the midpoint of the segment that has endpoints at (5,8) and (2,6).

1 5 forms of linear equations
1.5 Forms of Linear Equations

The slope intercept form of a line is y = mx + b. The slope is m and the y-intercept is b.

If the point with coordinates (x1,y1) lies on a line having slope m, the point-slope form can be written as follows:

The standard form of a linear equation is Ax + By + C = 0, where A, B, and C are real numbers and A and B are not both zero.

slide20

Example: Write the equation 4x – 3y + 7 = 0 in slope-intercept form. Then identify the slope and y-intercept.

Example: Write the slope-intercept form of the equation of the line through (3,7) that has a slope of 2.

Example: Find the equation of the line through (-2,4) and has a slope of -1.

1 6 parallel and perpendicular lines
1.6 Parallel and Perpendicular Lines

Two nonvertical lines are parallel if and only if their slopes are equal. Any two vertical lines are always parallel.

Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals.

slide23

Example: Write the standard form of the line that passes through (2,-3) and is parallel to the line 4x – y +3 = 0.

Example: Write the standard form of the line that passes through (3,-5) and is perpendicular to the line 2x – 3y + 6 = 0.