1.1 Functions and Their Representations

1.1 Functions and their representations

Notation: the set of real numbers

Integers are … , –3, –2, –1, 0, 1, 2, 3, …

Union

Intersection

Set builder notation: {x | x > 2}

Function f: A B (here A and B are subsets of real numbers)

f is a rule that assigns to each number in A exactly one number from B

A is the domain of f

f(x) is the value of f at x

range is the set of values in B that are obtained as output values of function

EXAMPLE A = [–1, 2] and B = and f: A B is given by f(x) = 3x – 2

domain:

range:

EXAMPLE For f(x) = 3x2 – 2x +1, find

1.  f(2 + h)

2.  f(x + h)

3. 

How can we tell if a graph is that of a function?

EXAMPLES

EXAMPLE Find domain and range for each function.

1.  f(x) = 3x – 2

2.  f(x) = x2

3.  f(x) =

4.  f(x) =

EQUATIONS OF LINES:

slope-intercept form point-slope form

y = mx + b y – y1 = m(x – x1)

slope is m and y-intercept is (0, b) slope is m and (x1, y1) is a point on the line

EXAMPLES

1.  Find the equation of the line through (1, –2) and (3, –1).

2.  Find the equation of the line through (2, 3) and parallel to 3x – y + 4 = 0.

Piecewise defined function

EXAMPLE Graph the function and find its domain and range. Also find f(16), f(2), and f(–100).

EXAMPLE A rectangle has area 16 m2. Express the perimeter of the rectangle as a function of the length of one of its sides.

EXAMPLE Express the area of a circle as a function of its circumference.

EXAMPLE Express the volume of a right circular cone with a vertex angle of 90o as a function of its height.

EXAMPLE An open box is made from a cardboard square that is 40 cm on a side by cutting out a square with length of a side x from each corner and then turning up the remaining sides. Express the volume of the box as a function of x.

EXAMPLE Water is being pumped into a conical tank whose dimensions are shown below. Express the volume of the water at any time t as a function of its depth h.

15