# 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 other functions such as trigonometric, inverse trigonometric... may be used as input function. Here is a comprehensive list of operators and functions 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^3-2*x^2-x+2

sin(2*x+1)

1-2*arctan(2*x)

(1-x)/(x^2-x-2)

NOTE that 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 calculator

1 - Press on the button "click here to start" to start the applet.

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2 - 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 black) 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).

Interactive Tutorial

## 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]

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