Download Sections 1 - UTEP Math

Sections 1.1, 1.2, 1.3
Functions
A real-valued function f of a real-valued variable x assigns to each real number x in a specified
set of numbers, called the domain of f, a unique real number f (x), read “f of x.” The variable x is
called the independent variable, and f is called the dependent variable.
The process of estimating values for a function between points where it is already known is called
interpolation. Estimating values for a function outside a range where it is already known is
called extrapolation.
Graph of a Function
The graph of the function f is the set of all points (x, f (x)) in the xy plane, where we restrict the
values of x to lie in the domain of f.
Vertical-Line Test
For a graph to be the graph of a function, every vertical line must intersect the graph in at most
one point.
Linear Function
A linear function is one that can be written in the form
f (x) = mx + b Function form or y = mx + b
where m and b are fixed numbers (some times we use other letters instead of m and b).
The Change in a Quantity: Delta Notation
If a quantity q changes from q1 to q2, the change in q is just the difference:
Change in q = Second value − First value = q2 − q1
Mathematicians traditionally use  (delta, the Greek equivalent of the Roman letter D) to stand
for change, and write the change in q as  q.
 q = Change in q = q2 − q1


The Roles of m and b in the Linear Function f(x) = mx +b

Role of m
Numerically If y = mx + b, then y changes by m units for every 1-unit change in x. A change of
 x units in x results in a change of  y = m  x units in y. Thus,
m

y Change in y

x Change in x
 of theline y = mx + b:
Graphically m is the slope
m
y Rise

 Slope
x Run
For positive m, the graph 
rises m units for every 1-unit move to the right, and rises  y = m  x
units for every  x units moved to the right. For negative m, the graph drops |m| units for every
1-unit move to the right, and drops |m|  x units for every  x units moved to the right.

Role
of b
Numerically When x = 0, y = b


Graphically b is the y-intercept of the line y = mx + b.


Finding the Intercepts
The x-intercept of a line is where it crosses the x-axis. To find it, set y = 0 and solve for
x. The y-intercept is where it crosses the y-axis. If the equation of the line is written in
as y = mx + b, then b is the y-intercept. Otherwise, set x = 0 and solve for y.
Computing the Slope of a Line
We can compute the slope m of the line through the points (x1, y1) and (x2, y2 ) using
m
A line of slope then it is a horizontal line.
y y 2  y1

x x 2  x1


The Point-Slope Formula 
An equation of the line through the point (x1, y1) with slope m is given by
y = mx + b (Equation form) where b  y1  mx1


Common Type of Algebraic Functions