Graph, Domain and Range of Common Functions
A tutorial using an HTML 5 applet to explore the graphical and analytical properties of some of the most common functions used in mathematics. The properties to be explored are: graphs, domain, range, interval(s) of increase or decrease, minimum or maximum and which functions are even, odd or neither . The answers to the questions in the tutorials are also included.
Functions to be Investigated
 linear function f(x) = x

squaring function g(x) = x^{2}

cubing function h(x) = x^{3}
 absolute value function i(x) = x
 square root function j(x) = √(x)

cube root function k(x) = ^{3}√(x)

natural exponential function l(x) = e^{x}
 natural logarithmic function m(x) = ln(x)
Interactive Tutorial Using Applet
TUTORIAL (1)  Domain and Range of Basic Functions
1  click on the button above "plot" to start.
2  Select a function and examine its graph. Write down its equation .(for example f(x) = x^{3}). Do this for all functions in the applet.
3  Domain : Select a function, examine its graph and its equation. Find the domain of the function being explored. Do this activity for all functions.
4  Range : Select a function, examine its graph and its equation. Find its range. Do this activity for all functions.
TUTORIAL (2)  Comparing Basic Functions
1  Select function f(x) = x and function i(x) =  x . Compare the two function and use the definition of the absolute value to explain how to graph i(x) =  x  from the graph of f(x) = x.
2  Compare the graphs of l(x) = e^{ x} and m(x) = ln (x). Why are the graphs reflection of each other on the line y = x?
3  Compare the graphs of h(x) = x^{ 3} and k(x) = ^{3}&radic (x). Why are the graphs reflection of each other on the line y = x?
4  Compare the graphs of f(x) = x, g(x) = x^{ 2}, h(x) = x^{ 3}, j(x) = √(x) and k(x) = ^{3}√(x). Write inequalities involving x, x^{ 2}, x^{ 3}, √(x) and ^{3}√(x) on the intervals 0 < x < 1 and x > 1?
TUTORIAL (3)  Intervals of Increase and Decrease and any local minimum or maximum of the Basic Functions
1  For each function, write the interval(s) where the function increases and the interval(s) where the function decreases and the coordinate of any local minimum or maximum.
TUTORIAL (4)  Compare the rate of change of the Basic Functions
1  Which of the functions g(x) = x^{2} or l(x) = e^{ x} increases faster as x increases?
2  Which of the functions f(x) = x or m(x) = ln x increases faster as x increases?
TUTORIAL (5)  Identify Even and Odd Basic Functions
1  An even function f has its graph symmetric with respect to the y axis and therefore satisfy the condistion f(x) = f(x). List all basic functions that are even.
2  An odd function f has its graph symmetric with respect to the origin of the system of axes and therefore satisfy the condistion f(x) =  f(x). List all basic functions that are odd.
3  List all basic functions that are neither even nor odd.
Answers 