The algebraic operations of functions are fundamental mathematical processes that combine functions to create new ones. According to Cuemath, these operations include addition, subtraction, multiplication, division, powers, and roots.
Understanding Algebraic Operations on Functions
These operations are similar to how we perform them on numbers, but instead of numbers, we're working with functions, which are expressions that define a relationship between variables.
The Operations
Here's a breakdown of each algebraic operation, along with examples:
| Operation | Notation | Description | Example |
|---|---|---|---|
| Addition | (f + g)(x) = f(x) + g(x) | The sum of two functions is found by adding their respective outputs for a given input. | If f(x) = x2 and g(x) = 2x, then (f + g)(x) = x2 + 2x |
| Subtraction | (f - g)(x) = f(x) - g(x) | The difference of two functions is found by subtracting the output of the second function from the output of the first function. | If f(x) = x2 and g(x) = 2x, then (f - g)(x) = x2 - 2x |
| Multiplication | (f g)(x) = f(x) g(x) | The product of two functions is found by multiplying their respective outputs for a given input. | If f(x) = x2 and g(x) = 2x, then (f g)(x) = x2 2x = 2x3 |
| Division | (f / g)(x) = f(x) / g(x), g(x)≠0 | The quotient of two functions is found by dividing the output of the first function by the output of the second function, where g(x) cannot equal zero. | If f(x) = x2 and g(x) = 2x, then (f / g)(x) = x2 / 2x = x/2, x≠0 |
| Powers | (f(x))n | Raising a function to a power involves raising the output of the function to that power. | If f(x) = x + 1, then (f(x))2 = (x + 1)2 = x2 + 2x + 1 |
| Roots | n√f(x) | Taking the root of a function involves finding the value that, when raised to the nth power, gives the output of the function. | If f(x) = x4, then √f(x) = √x4 = x2 |
Important Considerations
It's crucial to remember that certain types of functions are not considered algebraic. According to Cuemath, these include functions that contain:
- Logarithms (log, ln)
- Trigonometric functions (sin, cos, tan, etc.)
- Inverse trigonometric functions (arcsin, arccos, arctan, etc.)
- Variables in the exponent (e.g., 2x)
Any function involving these elements would be classified differently (e.g., as a transcendental function).
