Glossary
Additive Transformations
Transformations that involve adding or subtracting a constant to the input or output of a function, typically resulting in vertical or horizontal shifts.
Example:
If f(x) = x² and g(x) = x + 3, then f(g(x)) = (x+3)² represents a horizontal shift, which is a type of additive transformation.
Analytic Representation
Working with functions using their algebraic equations to perform operations like composition or decomposition.
Example:
To find f(g(x)) when f(x) = 2x + 1 and g(x) = x², you use analytic representation by substituting x² into f(x) to get 2(x²) + 1.
Composite Functions
A function formed by applying one function to the results of another function, essentially chaining them together.
Example:
If a function calculates the cost of ingredients for a cake, and another function calculates the selling price based on ingredient cost, then combining them to find the selling price directly from the ingredients is a composite function.
Function Decomposition
The process of breaking down a complex function into two or more simpler functions whose composition results in the original complex function.
Example:
If you have h(x) = (x+5)³, function decomposition might involve identifying f(x) = x³ and g(x) = x+5, so h(x) = f(g(x)).
Function Notation
The symbolic way to represent functions, especially composite functions, such as f(g(x)) or f ∘ g(x).
Example:
When you see f(g(x)), it tells you to evaluate the inner function g(x) first, then use that result as the input for the outer function f(x).
Graphical Methods (Composition)
Determining the output of a composite function by reading values from the graphs of the individual functions.
Example:
To find f(g(1)) using graphical methods, you locate x=1 on the graph of g(x) to find its y-value, then use that y-value as the x-input on the graph of f(x) to find the final output.
Horizontal Dilation
A multiplicative transformation that stretches or shrinks a function's graph horizontally by multiplying the input variable by a constant.
Example:
The function f(x) = sin(2x) represents a horizontal dilation (shrink) of the basic sine wave by a factor of 1/2.
Identity Function
A function, typically f(x) = x, that returns the same value as its input, leaving any function it's composed with unchanged.
Example:
If you have a function that doubles a number, and you compose it with the identity function, the number will still just be doubled, as the identity function doesn't alter its input.
Multiplicative Transformations
Transformations that involve multiplying the input or output of a function by a constant, leading to stretches, shrinks, or reflections.
Example:
If f(x) = sin(x) and g(x) = 2x, then f(g(x)) = sin(2x) represents a horizontal shrink, which is a multiplicative transformation.
Numerical Methods (Composition)
Evaluating composite functions by using specific numerical input values and following the chain of operations through tables or direct calculation.
Example:
To find f(g(3)) using numerical methods, you first find g(3) from a table, then use that output as the input for f(x) from its table.
Order of Operations (Composition)
Refers to the sequence in which functions are applied in a composition; it is crucial because f(g(x)) is generally not equal to g(f(x)).
Example:
If a function adds 2 and another multiplies by 3, applying 'add then multiply' (f(g(x))) gives a different result than 'multiply then add' (g(f(x))), showing that order of operations matters.
Vertical Translation
An additive transformation that shifts a function's graph up or down by adding or subtracting a constant to the function's output.
Example:
The function f(x) = x² + 5 is a vertical translation of f(x) = x² shifted 5 units up.