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Questions and Answers
What is the name of the set X in the context of functions?
What is the name of the set X in the context of functions?
Domain
Given a relation f: X →Y, what condition must hold for f(x) to be a function?
Given a relation f: X →Y, what condition must hold for f(x) to be a function?
For every x ∈ X, there must exist a unique y ∈ Y such that y = f(x).
The vertical line test is used to determine if a relation is a function graphically.
The vertical line test is used to determine if a relation is a function graphically.
True
Which of the following relations is a function?
Which of the following relations is a function?
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A function from a set D to a set Y is a ______ that assigns a unique element to each element in D.
A function from a set D to a set Y is a ______ that assigns a unique element to each element in D.
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The output of a function is also known as its range.
The output of a function is also known as its range.
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What is the horizontal line test used to determine?
What is the horizontal line test used to determine?
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Which of the following functions is one-to-one?
Which of the following functions is one-to-one?
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What is the inverse of a function?
What is the inverse of a function?
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A function and its inverse are always symmetrical about the line y = x.
A function and its inverse are always symmetrical about the line y = x.
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What is the first step typically involved in finding the inverse of a function?
What is the first step typically involved in finding the inverse of a function?
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How can you graphically find the inverse of a function?
How can you graphically find the inverse of a function?
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Study Notes
Functions
- A function maps elements from a set (domain) to a set (co-domain)
- A function assigns a unique output value for each input value
- Algebraically: for all x in the domain, there's a unique y in the co-domain such that y = f(x)
- Graphically: the vertical line test - a relation is a function if any vertical line intersects the graph at most once.
Example Functions
- f(x) = 2x + 1 is a function
- x² + y² = 9 is not a function
Domain and Range
- Domain: the set of all possible input values (x-values)
- Range: the set of all possible output values (y-values)
Examples of function Types and Domains
- y = x² Domain: (-∞, ∞) Range: [0, ∞)
- y = 1/x Domain: (-∞, 0) U (0, ∞) Range: (-∞, 0) U (0, ∞)
- y = √x Domain: [0, ∞) Range: [0, ∞)
- y = √(4 - x) Domain: (-∞, 4] Range: [0, ∞)
- y = √(1 - x²) Domain: [-1, 1] Range: [0, 1]
One-to-One Functions
- A function f(x) is one-to-one if f(x₁) ≠ f(x₂) whenever x₁ ≠ x₂
- Algebraically: f(x₁) = f(x₂) implies x₁ = x₂
- Graphically: the horizontal line test - a function is one-to-one if any horizontal line intersects the graph at most once.
Inverse Functions
- An inverse function reverses the relationship of a one-to-one function
- g(x) is the inverse of f(x) if g(f(x)) = x for all x in the domain of f(x) and f(g(x))=x for all x in the domain of g(x).
- Notation: g(x) is often written as f⁻¹(x).
- f⁻¹(f(x)) = x and f(f⁻¹(y)) = y
- To find the inverse, solve for x in terms of y, then swap x and y.
- Example: If y = (1/2)x + 1, f⁻¹(x) = 2x - 2
How to Sketch the Inverse
- Draw the line y = x
- Reflect the graph of f over the line y = x. The reflection is f⁻¹(x)
Finding an Inverse Function
- Given a function, replace y with f(x). Solve for x in terms of y. Swap x and y. Replace y with f⁻¹(x).
- Example: if y=√x then x = y^2 and f^(-1) (x)=x^2.
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Description
Explore the concepts of functions, including their definitions, domains, and ranges. This quiz will cover various types of functions and the vertical line test to determine function validity. Test your knowledge on one-to-one functions and example functions provided.
