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# Operations on Functions - Graphing Calculator

An online graphing calculator to carry out operations on functions. Five operations are supported by this calculator: addition, subtraction, multiplication, division and composition. . The calculator has two inputs: one for function f and a second one for function g. Algebraic as well as trigonometric, inverse trigonometric, exponential , logarithmic and hyperbolic functions may be used as input function. Here is a comprehensive list of basic functions and operators that may be used.

Examples of formulas for functions f and g, that you may copy and paste to use as inputs, are shown below:
x - 2
- 2 x^2 + 3 x -1
sin(x+pi)
sin(pi*x)
atan(2*x)
(1-x)/(x^2-x-2)
sinh(x-1)

NOTE: The multiplication operator * must be used explicitly whenever there is multiplication. 2sin(x) will not be accepted. It must be written 2*sin(x). Also the argument of a function must be writtwn between brackets. sin x is not accepted. It must be written sin(x).
If needed, Free graph paper is available.

## How to Use The Operations on Functions Calculator

Enter formulas for functions f and g and press the button corresponding to the operation to be carried out on functions f and g and explore the graphs of the three functions: f (in blue), g (in green) and the graph of function due to the operation carried out on f and g (in red). Five operations are supported by this calculator. (see more details on each operation below). Use the small letter x for the variable in the expressions of functions f and g .

f(x) =       g(x) =

Hover the mousse cursor over the graph to trace the coordinates.
Hover the mousse cursor on the top right of the graph to have the option of downloading the graph as a png file, zooming in and out, shifting the graphs, ....

## Interactive Tutorial

Some tutorials and activities are suggested here but the use of this graphing calculator to explore and gain deep understanding of operations on functions is unlimited and any suggestions are welcome.

### 1 - Addition of two functions

• Let f(x) = sqrt(x) and g(x) = x, input functions f and g and press on the button "(f + g)(x)".
• Explore the graph (in red) of function (f + g)(x); take some specific values of x if necessary; at x = 1 for example. Find f(1), g(1) and check that (f+g)(1)=f(1)+ g(1) (this is the definition).
• Explore the domain of f + g graphically. Is it the intersection of the domains of f and g?

### 2 - Subtraction of two functions

• Let f(x) = x + 1 and g(x) = x, input functions f and g and press on the button "(f - g)(x)".
• Explore the graph (in red) of function (f - g)(x); is it what is expected?
• Let f(x) = sqrt(x-2) and g(x) = x, input functions f and g and press on the button "(f - g)(x)". Check that at x = 2, (f - g)(2) = f(2) - g(2). Do the same at x = 3 and some other points. Explore the domain of f - g graphically. Is it the intersection of the domains of f and g?

### 3 - Multiplication of two functions

• Let f(x) = x - 2 and g(x) = x, input functions f and g and press on the button "(f * g)(x)". Explore the graph (in red) of function (f * g)(x); is it what is expected? Compare the zeros of f, g and f * g and explain.
• Let f(x) = sqrt(x + 2) and g(x) = x, input functions f and g and press on the button "(f * g)(x)". Check that at x = -2, (f * g)(2) = f(2) * g(2). Do the same at x = -1 and some other points. Explore the domain of f * g graphically. Is it the intersection of the domains of f and g?

### 4 - Division of two functions

• Let f(x) = 1 and g(x) = x, input functions f and g and press on the button "(f / g)(x)". Explore the graph (in red) of function f / g; is it what is expected? What do you think is happening at x = 0 for the graph of f / g?
• Let f(x) = sqrt(x) and g(x) = x - 1, input functions f and g and press on the button "(f / g)(x)". Check that at x = 0, (f / g)(0) = f(0) / g(0). Do the same at x = 4 and some other points. Explore the domain of f / g graphically. Is it the intersection of the domains of f and g? Why is (f/g)(x) undefined at x = 1? What is the domain of f / g?

### 5 - Composition of two functions

• Let f(x) = sqrt(x) and g(x) = x^2, input functions f and g and press on the button "(f o g)(x)". Estimate (f o g)(2) from the graph (red). Estimate g(2) from graph (black) and now estimate f(g(2)) from graph (blue). (f o g)(2) and f(g(2)) should be very close if not equal. Do the same for (f o g)(4) and f(g(4)).
• Explore the domain of (f o g)(x).
• Definition: The domain of (f o g) is the set of all values of x such that g(x) is defined and real and also f(g(x)) is defined and real. Is the domain of (f o g) what is expected? Explain.

## Exercises

1 - Let f(x) = sqrt(1-x) and g(x) = x^2. Input functions f and g and press on the button "(f o g)(x)". Estimate the domain of (f o g) from graph? Determine the domain analytically and compare.
2 - Let f(x) = 1 - x and g(x) = sqrt(x). Input functions f and g and press on the button "(f - g)(x)". Using the graph, what do you think is the domain of f - g? Explain analytically.
3 - Let f(x) = 1 and g(x) = sqrt(x). Input functions f and g and press on the button "(f / g)(x)". Using the graph, what do you think is the domain of f / g? Explain analytically.
4 - Let f(x) = sqrt(1-x^2) and g(x) = sqrt(4-x^2). Input functions f and g and press on the button "(f * g)(x)". Estimate the domain of (f o g) from graph? Determine the domain analytically and compare.
Solutions to above exercises:
1 - Since the domain of g is the set of all real numbers, the domain of (f o g)(x) is all values of x such that 1 - g(x) >= 0 or 1 - x^2 >= 0. Solving the inequality we obtain the domain as the interval: [-1 , 1]
2 - The domain of f - g is the intersection of the domain of f which is the set of all real numbers and the domain of g which is the set [0 , + infinity). The intersection is given by the interval [0 , + infinity).
3 - The domain of f / g is the intersection of the domain of f which is the set of all real numbers and the domain of g which is the set [0 , + infinity) also excluding any values of x that make the denominator of (f/g)(x) equal to zero (division by zero is not allowed). The intersection is given by the interval [0 , + infinity), exclude x = 0 since g(0) = 0, the final domain is given by (0,+infinity).
4 - The domain of f is the set of values in [-1,1] and the domain of g is the set of values in [-2,2]. The intersection of the two sets is [-1,1]