Function Operations Review

This cheat sheet covers the fundamental operations of functions, including addition, subtraction, multiplication, division, and composition, along with domain and range considerations. It provides a concise review of key concepts and problem-solving techniques.

Core Principles

Function Addition: (f + g)(x) = f(x) + g(x)
Function Subtraction: (f - g)(x) = f(x) - g(x)
Function Multiplication: (f * g)(x) = f(x) * g(x)
Function Division: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
Function Composition: (f ∘ g)(x) = f(g(x))
Domain of combined functions: Consider restrictions from original functions and new denominators.
Range of combined functions: Determined by analyzing the output of the combined function.

Action Steps

Identify the type of function operation required.
Substitute the function expressions into the operation formula.
Simplify the resulting expression.
Determine the domain by considering restrictions from the original functions and any new denominators.
Determine the range by analyzing the behavior of the combined function, often by sketching its graph.

Formulas

(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f \cdot g)(x) = f(x) \cdot g(x)
(\frac{f}{g})(x) = \frac{f(x)}{g(x)}, \quad g(x) \neq 0
(f \circ g)(x) = f(g(x))

Key Terms

Function Addition: Combining two functions by adding their outputs for each input value.
Function Subtraction: Combining two functions by subtracting the second function's output from the first for each input value.
Function Multiplication: Combining two functions by multiplying their outputs for each input value.
Function Division: Combining two functions by dividing the first function's output by the second, ensuring the divisor is not zero.
Function Composition: Applying one function to the result of another function.
Domain: The set of all possible input values (x-values) for which the function is defined.
Range: The set of all possible output values (y-values) that the function can produce.

Real World Examples

Sales data from two store locations over time.: Combining sales functions to find total sales or analyze trends.
Cost of production as a function of quantity, and quantity as a function of hours.: Composing functions to find the cost of production directly in terms of hours.