# 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

1. linear function f(x) = x
2. squaring function g(x) = x2
3. cubing function h(x) = x3
4. absolute value function i(x) = |x|
5. square root function j(x) = √(x)
6. cube root function k(x) = 3√(x)
7. natural exponential function l(x) = ex
8. natural logarithmic function m(x) = ln(x)

## Interactive Tutorial Using Applet

f(x) = x g(x)= x 2 h(x) = x 3 i(x) = | x | j(x) = √(x) k(x) = 3√(x) l(x) = ex m(x) = ln x

## 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) = x2 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.