Download Note 1. Functions. A function is a rule that assigns to each object in a

Note 1.
Functions.
A function is a rule that assigns to each object in a set A exactly one object in a set B.
The set A is called the domain of the function.
The set B is called the range of the function.
The natural domain of f is the set of all numbers for which f (x) is defined.
A function can be written as
y = f (x)
x and y are called variables.
x is called the independent variable,
and y is called the dependent variable.
A piecewise-defined function is a function that is defined using more than one formula.
In many situations, a quantity y is given as a function f of one variable u (we write this as y = f (u))
that, in turns, can be written as a function g of a second variable x (written u = g(x)). In that case we
have composition of function, namely y can be written as the composition of f and g as y = f (g(x)).
Be careful about the ordering, since f (g(x)) is in general different from g( f (x)).
Functions used in economics.
The demand function D(x) for a commodity is the price p = D(x) that must be charged for each unit of
the commodity if x units are to be sold.
The supply function S(x) for a commodity is the unit price p = S(x) at which producers are willing to
supply x units to the marker.
The revenue function R(x) is obtained from selling x units of the commodity, and is given by the product
R(x) = x p(x)
The cost function C(x) is the cost of producing x units of the commodity.
The profit function P(x) is the profit obtained from selling x unit of the commodity and is given by the
difference
P(x) = R(x) −C(x) = xp(x) −C(x)
We sometimes call x the production level.
The graph of a function
A point in the plane, say P, is a unique ordered pair of numbers (a, b). In this course, we will often
write this as P(a, b).
- (a, b) are the coordinates of P.
- a is the x coordinate,
- b is the y coordinate.
The distance between P(x1 , y1 ) and Q(x2 , y2 ) is given by
q
D = (x2 − x1 )2 + (y2 − y1 )2
The graph of a function is the set of all the points of the form (x, f (x)). We can draw the graph by:
- Plotting a few points and connect the dots using a smooth curve
Pros: easy. Cons: not accurate.
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- Using analytic tool, which we will study in Chapter 3.
The vertical line test: a curve is the graph of a function if and only if no vertical line intersects the curve
more than once.
Intercepts are points where a graph crosses any of the axes.
The point where a graph crosses the x-axis is called x-intercepts. Set y = 0 and find x.
The point where a graph crosses the y-axis is called y-intercept. It is of the form (0, f (0)).
Graphing lines
Lines are graphs of linear functions: y = mx + b, with A 6= 0 (we will learn more about linear function in
Section 1.3).
Graphing parabolas
Parabolas are graphs of quadratic functions: y = Ax2 + Bx +C, with A 6= 0. It has ’U’ shape, opens up if
B
A > 0 and opens down if A < 0. The peak or valley is called vertex, occurs where x = − .
2A
Other functions
Power functions are of the form f (x) = xn , n is a real number.
Polynomial are of the form f (x) = an xn + an−1 xn−1 + ... + a1 x + a0 . If an 6= 0 we say n is the degree of
the polynomial.
Rational functions is the quotient of 2 polynomials.
Linear functions
Linear functions can be written as f (x) = mx + b.
∆y
The slope of a line is m = .
∆x
It can be written in the slope-intercept form ( f (x) = mx + b)
or in the point-slope form ((y − y0 ) = m(x − x0 )).
Two lines are parallel if m1 = m2 , and perpendicular if m2 =
−1
.
m1
Functional models
Model real-life problems using functions, and answer some of the most sought-after-questions (minimizing cost, maximizing profit,...).
Terms used in this:
- Directly proportional, inversely proportional, jointly proportional.
- Law of supply and demand:
the market is in equilibrium if supply equal demand,
equilibrium price is the price at which equilibrium occurs,
the market has a shortage if demand exceeds supply, it has a surplus if supply exceeds demand.
- Break-even analysis:
break-even point: the point where there is no loss, no profit.
Basic terminologies
- A quadratic equation is an equation of the form Ax2 + Bx +C = 0 with A, B,C constants.
- A quadratic function f is a function of the form f (x) = Ax2 + Bx +C with A, B,C constants.
- The quadratic formula is:
√
−B − B2 − 4AC
2
If B − 4AC > 0, the quadratic equation has two distinct solutions x1 =
and x2 =
2A
2
√
−B + B2 − 4AC
.
2A
In the particular case where B2 − 4AC = 0, the two solutions above coincide. We say that the quadratic
B
equation has one solution x = − .
2A
Another situation is, if B2 − 4AC < 0, the quadratic equation has no solution.
- For a simple quadratic equation, you can also guess the solutions and write it as A(x − x1 )(x − x2 ) = 0.
Explanation on factoring a quadratic formula There are two ways to write f (x) = Ax2 + Bx + C as a
product of the form f (x) = A(x − x1 )(x − x2 ).
First way: Factor out the value of A: f (x) = A x2 + BA x + CA then use the guessing method.
√
−B − B2 − 4AC
2
Second way: Use quadratic formula: First check that B − 4AC > 0. Then find x1 =
2A
√
−B + B2 − 4AC
and x2 =
and write f (x) = A(x − x1 )(x − x2 ).
2A
Now you have the two numbers to put on the line for the sign check.
Some basic techniques you need:
- The order of calculation operations: parenthesis, multiplication/division, addition/subtraction.
- Addition and subtraction are used to cancel each other out.
Multiplication and division are used to cancel each other out.
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