# Equations

# Equations, functions and graphs

**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} − 2xy + 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
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}.

**Arrow diagram of f**

**Machine diagram of f**

**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} − 3x + 1 can be thought of as

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 = e^{x} + x^{3} − 3x + 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:

**Example 5:
**write an expression for:

**Example 6**

If P(x) = 3x^{3}+ 2x^{2} + x + 11 and we
rewrite P(x) in the form

P(x) = A + B(x − 1) + C(x − 1)(x − 2) + D(x − 1)(x − 2)(x − 3),

what are the values of A and B?

**Example 7:**

If we rewrite the function
in the form:

what are the values of A, B, and C?

Inverse of a function:

Recall that two functions f(x) and g(x) are said to be
inverse of each other if

f(g(x)) = x and g(f(x)) = x.

**Example 8:**

If h(t) = 3t + 7, find a function g(t) such that h(g(t)) = t.

**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.

**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).

**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:

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.

Graph of a function

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 ≠ 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.

f(x) = a(x − h)^{2} + k,

a > 0

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}−15x+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} − 15x + 54 < 0 in the form A < x < B,
what are the values of A and B?

Intersection points:

The graphs of two equations intersect at a point if and only if that point is a solution for both equations.

**Example 11:**

Find the point(s) of intersection between the graph of the
equation 4x^{2} + 9y^{2} = 36 and

the line with equation

• y = −2;

• y = 1.

**Example 12:**

Find all points where the graph of (y − 1)^{2} − 2
= x crosses

• the y-axis;

• the x-axis.

**Example 13:**

The area of a right triangle is 7. The sum of the lengths
of the two sides adjacent to the

right angle of the triangle is 11. What is the length of the hypotenuse of the
triangle?

**Example 14:**

What is the smallest root of the polynomial Q(x) = x^{3}
− 12x^{2} + 44x − 48?

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