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MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date:
Chapter Goals: •^ Identify solutions to an equation.
- Solve an equation for one variable in terms of another.
- What is a function?
- Understand function notation.
- Find inverse functions.
- What is a graph?
- Sketch the graph of an equation in two variables.
- Find the equation of a line and a parabola.
- Find the intersection point(s) of two graphs. Assignments: (^) Assignment 00 (Due date: Jan. 20) Assignment 01 (Due date: Jan. 22)
◮ Equations and solution(s) to equations: One way in which humanity increases its understanding of the universe is by discovering relationships between various objects, concepts, quantities, and so on. Our understanding of a relationship between two quantities is sharpest when this relationship can be completely quantified and expressed in an equation. Roughly speaking, an equation is a statement that two mathematical expressions are equal. For instance, x^3 − 2 xy + y^2 = 5 is an equation relating x and y. A set of numbers that can be substituted for the variables in an equation so that the equality is true is a solution for the equation. A solution is said to satisfy the equation.
Example 1:
Is x = 1 and y = 2 a solution for the equation x^2 + y = 2xy? What about x = 1 and y = 1?
Many problems in the sciences, economics, finance, medicine and numerous other fields can be formulated into algebraic terms by identifying variables expressing unknown quantities and by setting up appropriate equations relating these variables.
Example 2: Suppose a fuel mixture is 4% ethanol and 96% gasoline. How much ethanol (in gallons) must you add to one gallon of fuel so that the new fuel mixture is 10% ethanol?
◮ Equations into functions: An equation in two (or more) variables can sometimes be solved in terms of one of the variables. This type of equation is closely related to the notion of a function.
Example 3: Solve the equation x^3 + 2xy + 5y = 7 for y in terms of x.
Observe that in the equation y =^7 −^ x
3 2 x + 5 , the expression on the right-hand side can be viewed as a recipe that associates to any given value of x precisely one corresponding value for y.
Definition of function: A function f is a rule that assigns to each element x in a set A exactly one element, called f (x), in a set B. The set A is called the domain of f whereas the set B is called the codomain of f ; f (x) is called the value of f at x, or the image of x under f. The range of f is the set of all possible values of f (x) as x varies throughout the domain: range of f = {f (x) | x ∈ A}.
x input
f (x) output
f
Machine diagram of f
Arrow diagram of f
f
A
x•
a
B
f (x)
f (a) = f (b)
Evaluating a function: The symbol that represents an arbitrary number in the domain of a function f is called an independent variable. The symbol that represents a number in the range of f is called a dependent variable. In the definition of a function the independent variable plays the role of a “placeholder”. For example, the function f (x) = 2x^2 − 3 x + 1 can be thought of as
f () = 2 · ^2 − 3 · + 1.
To evaluate f at a number (expression), we substitute the number (expression) for the placeholder.
Note: If f is a function of x, then y = f (x) is a special kind of equation, in which the variable y appears alone on the left side of the equal sign and the expression on the right side of the equal sign involves only the other variable x. Conversely, when we have this special kind of equation, such as y = ex^ + x^3 − 3 x + 5, it is common to think of the right hand side as defining a function f (x), and of the equation as being simply y = f (x).
Example 4: Find the domain of the following functions:
f (x) =
3 − x g(x) = (^) x (^2 1) − 4 h(x) = x^1 +
x + 2
◮ Cartesian plane and the graph of a function:
Points in a plane can be identified with ordered pairs of numbers to form the coordinate plane. To do this, we draw two perpendicular oriented lines (one horizontal and the other vertical) that intersect at 0 on each line. The horizontal line with positive direction to the right is called the x-axis; the other line with positive direction upward is called the y-axis. The point of intersection of the two axes is the origin O. The two axes divide the plane into four quadrants, labeled I, II, III, and IV. The coordinate plane is also called Cartesian plane in honor of the French mathematician/philosopher Ren´e Descartes (1596-1650). Any point P in the coordinate plane can be located by a unique ordered pair of numbers (a, b) as shown in the picture. The first number a is called the x-coordinate of P ; the second number b is called the y-coordinate of P.
x
y
O
a
b
II I
III IV
Graphing functions:
If f is a function with domain A, then the graph of f is the set of ordered pairs
graph of f = {(x, f (x)) | x ∈ A}.
In other words, the graph of f is the set of all points (x, y) such that y = f (x); that is, the graph of f is the graph of the equation y = f (x). 0
x
y
x
f (x)
(x, f (x))
f (6) (6, f^ (6))
f (2) (2, f (2))
Obtaining information from the graph of a function:
The values of a function are represented by the height of its graph above the x-axis. So, we can read off the values of a function from its graph. In addition, the graph of a function helps us picture the domain and range of the function on the x-axis and y-axis as shown in the picture: (^0)
x
y
Range y = f (x)
Domain
The graph of a function is a curve in the xy- plane. But the question arises: Which curves in the xy-plane are graphs of functions?
The vertical line test: A curve in the coordinate plane is the graph of a function if and only if no vertical line intersects the curve more than once. x
y
Graph of a function
x
y
Not a graph of a function
◮ Lines and parabolas: The simplest types of functions are linear and quadratic functions.
A linear function is a function f of the form f (x) = mx + b, where m and b are real numbers. The graph of the equation y = mx + b is a (non-vertical) line in the xy-plane. The numbers m and b are called the slope and y-intercept, respectively.
A quadratic function is a function f of the form
f (x) = ax^2 + bx + c,
where a, b, and c are real numbers and a 6 = 0. The graph of any quadratic function is a parabola; it can be obtained from the graph of f (x) = x^2 by using shifting, reflecting and stretching trans- formations. Indeed, by completing the square a quadratic function f (x) = ax^2 + bx + c can be expressed in the standard form
f (x) = a(x − h)^2 + k.
The graph of f is a parabola with vertex (h, k); the parabola opens upward if a > 0, or downward if a < 0.
x
y
h
k (^) Vertex (h, k) (Minimum)
f (x) = a(x − h)^2 + k, a > 0
x
y
h
k Vertex^ (h, k)
(Maximum)
f (x) = a(x − h)^2 + k, a < 0
Example 9: If the equation of the line through the points (3, 4) and (− 1 , 6) is written as
y = A + B(x + 1),
what are the values of A and B?
Example 10: The parabola y = x^2 − 15 x + 54 intersects the x-axis at the two points P and Q. What is the distance from P to Q? If we rewrite the inequality x^2 − 15 x + 54 < 0 in the form A < x < B, what are the values of A and B?
