Linear Functions

Definition A function f is linear if its domain is a set of numbers and it can be expressed in the form where m and b are constants and x denotes an arbitrary element of the domain of f.

Change and Rate of Change

Definition If x1 and x2 are distinct members of the domain of f, the change in f from x1 to x2 is f(x2) – f(x1). The rate of change of f over the interval from x1 to x2 is

Notation Let Dx = x2 – x1 denote the change in x. Let Df =f(x2) – f(x1) denote the change in f. The rate of change is the ratio

Exercise For real numbers x, let . Find the change in f from x1 = 1 to x2 = 4. Find the rate of change of f over the interval from 0 to 3 . Find a general formula for the rate of change over the interval from x1 to x2 for any x1 and x2.

A Characterization of Linear Functions A function from the real numbers to the real numbers is linear if and only if its rate of change is the same for all intervals. If so, the rate of change is the constant m in the formula

Graphs of Linear Functions Straight Lines

Two distinct points in the plane determine one and only one straight line

Point-Slope Form Let be two distinct points in the plane. Case 1: Set (slope) Equation: or

Case 2: Equation: x = c.

Point-Slope Form Suppose it is known that a line passes through the point with coordinates and that it has slope m. Then the equation of the line is

Slope Intercept Form y = f(x) = mx + b m = rate of change of f = slope of the line = tangent of angle between the x-axis and the line b = f(0) = y-intercept of the line

Geometrical Interpretation

The Symmetric Form Slope-intercept and point-slope forms cannot handle vertical lines in the xy plane. Symmetric form does not select one variable as the independent variable and the other as the dependent variable. c, d, and e are constants.

Exercise The graph of a linear function is the line whose equation is What is the rate of change of f? What are f(0) and f(-2)?

Systems of Linear Equations

General Form of a Linear System of Two Equations in Two Unknowns Equations in Symmetric Form of Two Straight Lines

Three Possibilities for Solutions The lines are not parallel and intersect in one and only one point. That is, there is one and only one solution of the system. The lines are distinct but parallel and do not intersect. There are no solutions. The equations represent the same straight line. There are infinitely many solutions, one for each point on the line.

Examples: 1. 2. 3.

The Coefficient Matrix

The Determinant of the Coefficient Matrix The number

Relationship of the Determinant to the Question of Solutions The linear system has a unique solution if and only if the determinant is different from zero.

Cramer’s Rule Not necessarily the best method of solution.

Exercise Solve Answer: x=3/7, y=2/7

Inverses of Linear Functions

Example Given y, solve for x:

Example (continued) The equation defines x as a linear function of y. This function is called the inverse of the original function. We write

Equivalence The two equations and are equivalent. One is satisfied by a pair (x,y) if and only if the other is.

General Expression for the Inverse Function If f (x) = mx + b and m≠0, then Note: The slope of the inverse function is the reciprocal of the slope of the original function.

The Graphs of the Function and Its Inverse