Questions and Answers
What is the composition of functions?
Process of combining two functions where one function is performed first and the result is substituted in the other function.
What is the inverse of f(x)=2x+3?
f^-1(x)=(x-3)/2
What is the inverse of f(x)=x?
f^-1(x)=x
What are inverse functions?
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What is a function?
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What is a domain in the context of functions?
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What is a range of a function?
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What are the inverse points of the set {(1, -3), (-2, 3), (5, 1), (6, 4)}?
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What are the inverse points of the set {(-5, 7), (-6, -8), (1, -2), (10, 3)}?
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Express the piecewise function f(x) = x if x > 0, = -x if x <= 0.
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Study Notes
Functions and Their Properties
- Composition of Functions involves combining two functions where the output of the first function becomes the input for the second function.
- A function is a relation that pairs each element of the domain precisely with one element of the range.
Inverse Functions
- The inverse of the function f(x) = 2x + 3 is f^(-1)(x) = (x - 3) / 2.
- The inverse of the function f(x) = x is f^(-1)(x) = x, indicating that it is its own inverse.
- For a function f(x), the inverse function f^(-1)(x) is verified if f(f^(-1)(x)) = f^(-1)(f(x)) = x, achieved by swapping input and output variables in the function.
Domains and Ranges
- The domain of a function is defined as the complete set of possible x-coordinates from its ordered pairs.
- The range refers to all possible output values (y-values) that can result from the function’s inputs.
Finding Inverses of Point Sets
- For the set {(1, -3), (-2, 3), (5, 1), (6, 4)}, the inverses are calculated as {(-3, 1), (3, -2), (1, 5), (4, 6)} by swapping each pair.
- For the set {(-5, 7), (-6, -8), (1, -2), (10, 3)}, the inverses obtained are {(7, -5), (-8, -6), (-2, 1), (3, 10)} by interchanging the coordinates.
Piecewise Functions
- The piecewise function is defined by:
- f(x) = x for x > 0
- f(x) = -x for x ≤ 0
- This definition demonstrates how the function's behavior varies depending on the value of x.
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