Operations on Functions Calculator

- May 09, 2025
| |

Calculate and visualise the results of various operations on functions including addition, subtraction, multiplication, division, and composition.

You can use standard functions like sin(x), cos(x), tan(x), sqrt(x), log(x), exp(x), abs(x), as well as constants like pi.

Function Definitions

Operation Selection

Visualisation Options

Understanding Operations on Functions

Operations on functions allow us to create new functions by combining existing ones. These operations follow specific rules and have important applications in mathematics and various scientific fields.

Basic Function Operations
  • Addition (f+g)(x) = f(x) + g(x): The sum of the function outputs at each input x.
  • Subtraction (f-g)(x) = f(x) - g(x): The difference of the function outputs at each input x.
  • Multiplication (f×g)(x) = f(x) × g(x): The product of the function outputs at each input x.
  • Division (f÷g)(x) = f(x) ÷ g(x): The quotient of the function outputs at each input x, defined when g(x) ≠ 0.
Function Composition

Function composition creates a new function by applying one function to the output of another:

  • (f∘g)(x) = f(g(x)): Apply g first, then apply f to the result.
  • (g∘f)(x) = g(f(x)): Apply f first, then apply g to the result.

Note that function composition is not commutative, meaning (f∘g)(x) is generally not equal to (g∘f)(x).

Domain Considerations

The domain of the resulting function is affected by the operation:

  • For addition, subtraction, and multiplication, the domain is the intersection of the domains of f and g.
  • For division, the domain is the intersection of the domains of f and g, excluding values where g(x) = 0.
  • For (f∘g)(x), the domain includes values of x where g(x) is in the domain of f.

Understanding the Operations on Functions Calculator

The Operations on Functions Calculator is a handy tool for anyone looking to perform mathematical functions with ease. This calculator allows users to conduct a variety of operations, such as addition, subtraction, multiplication, division, and composition of functions. Whether you're a student or just need a quick calculation, this tool simplifies the process of working with functions.

How to Define Your Functions

When you start using the calculator, you’ll need to define the functions you want to work with. You can input standard functions like sin(x), cos(x), and sqrt(x). Here are some examples of how to define your functions:

  • Function f(x): You might input something like x^2 or log(x).
  • Function g(x): Common examples include x + 1 or 3*x - 2.

Selecting the Right Operation

Choosing the right operation is crucial for getting the results you need. The calculator provides a dropdown menu where you can select the operation you want to perform. You can choose from:

  • Addition (f + g)
  • Subtraction (f - g)
  • Multiplication (f × g)
  • Division (f ÷ g)
  • Composition (f ∘ g or g ∘ f)

Visualising Your Function Results

This calculator doesn’t just give you numbers; it also provides a visual representation of the functions. After performing the calculations, you can see the graphs of your functions on the same screen. You can adjust the x-axis limits to focus on different parts of the graph, ensuring you get a complete view of how the functions behave.

Finding Evaluations at Specific Points

An exciting feature of this calculator is the ability to evaluate functions at specific points. You can input a value for x and find out the results of both individual functions and their combination. For instance, after defining your functions, you can evaluate f(2) and g(2) to see what they equal when x = 2.

Understanding Domain Considerations

The domain of a function is the set of all possible input values. Different operations can affect the domain of the resulting function. For example:

  • For addition and subtraction, the domain is the overlap of the domains of both functions.
  • For division, you must exclude values where the denominator is zero.
  • For composition, ensure the output of the inner function falls within the domain of the outer function.

Learning About Basic Function Operations

Getting familiar with basic operations is essential for using the calculator effectively. Here’s a quick rundown of the primary operations:

  • Addition: \( (f+g)(x) = f(x) + g(x) \)
  • Subtraction: \( (f-g)(x) = f(x) - g(x) \)
  • Multiplication: \( (f×g)(x) = f(x) × g(x) \)
  • Division: \( (f÷g)(x) = f(x) ÷ g(x) \) when \( g(x) eq 0 \)

Wrapping Up with function composition

Function composition is an exciting aspect of working with functions. By combining functions, you can create new ones. Remember that the order matters in composition. For instance, \( (f∘g)(x) \) applies g first, then f. This is a powerful feature that can lead to new insights and discoveries in your calculations!