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All Flashcards
Explain the order of operations in .
Evaluate first, then use the result as the input for . Work from the inside out.
Why is the order important in composite functions?
Composition is generally not commutative; is usually not equal to .
What happens when you compose a function with the identity function?
The result is the original function; implies .
How do you find analytically?
Substitute the entire function for every instance of in .
How do composite functions relate to transformations?
They can represent transformations such as vertical translations () and horizontal dilations ().
Explain how to use graphs to evaluate composite functions.
Find the output of from its graph, then use that output as the input for on its graph.
What is a composite function?
A function formed by applying one function to the results of another: .
What does mean?
Apply to first, then apply to the result.
What is the identity function?
The function , which returns the input unchanged.
Define function decomposition.
Breaking down a complex function into simpler component functions.
What is vertical translation in terms of function composition?
Shifting the graph of a function up or down by adding a constant, represented by .
What is horizontal dilation in terms of function composition?
Stretching or shrinking the graph of a function horizontally by multiplying the input by a constant, represented by .
How to find given and ?
Replace in with : .
How to evaluate given and ?
First, find . Then, find .
How to decompose into two functions, and ?
Let and . Then .
How to find such that , given and ?
First find . Then solve 2x + 1 = 5
, which gives .
How to find if and ?
Substitute into : , for .
How to determine the domain of ?
Find the domain of and ensure that the range of is within the domain of .