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. A function is a relation in which each element of the domain is paired with exactly one element in the range.

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:

A function is usually denoted by f 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?

Find the domain of each of the following functions.

2.1 Compositions and Inverses of Functions Operations with functions: Sum Difference Product Quotient

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

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.

If and , find Compose onto itself, this is called an iteration.

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.

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

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.

Graph the following

Graph the following

Graph the following

1.4 Distance and Slope We will now derive the distance formula.

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).

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 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.

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

Example: Write the equation of the line passing through (3,7) and (4,-1).

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.

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.