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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 (A)
Which of the following relations is a function?
Which of the following relations is a function?
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.
The output of a function is also known as its range.
The output of a function is also known as its range.
What is the horizontal line test used to determine?
What is the horizontal line test used to determine?
Which of the following functions is one-to-one?
Which of the following functions is one-to-one?
What is the inverse of a function?
What is the inverse of a function?
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.
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?
How can you graphically find the inverse of a function?
How can you graphically find the inverse of a function?
Flashcards
Function
Function
A rule that assigns a unique element from a set (Y) to each element of another set (X).
Vertical Line Test
Vertical Line Test
A graphical method to determine if a graph represents a function. A graph is a function if no vertical line intersects the graph more than once.
Domain
Domain
The set of all possible input values (x) for a function.
Range
Range
The set of all possible output values (y) for a function.
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One-to-One Function
One-to-One Function
A function where each input value results in a unique output value; and distinct input values have distinct output values.
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Horizontal Line Test
Horizontal Line Test
A graphical method to determine if a function is one-to-one. A function is one-to-one if no horizontal line intersects the graph more than once.
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Inverse Function
Inverse Function
A function that reverses the action of another function, undoing the operation.
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f⁻¹(x)
f⁻¹(x)
Notation for the inverse function of f(x).
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f(x)=2x+1
f(x)=2x+1
A linear function with a slope of 2 and a y-intercept of 1.
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y=x²
y=x²
A quadratic function that produces a parabola.
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y=x³
y=x³
A cubic function that produces an S-shaped curve.
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Finding the inverse of a function
Finding the inverse of a function
Finding the function that reverses the operation of the original function.
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f(x) = √x
f(x) = √x
A square root function with a domain of x ≥ 0 and range of y ≥ 0.
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f(x) = 1/x
f(x) = 1/x
A reciprocal function with domain excluding zero and range excluding zero.
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Graph Symmetry
Graph Symmetry
A visual relationship between two graphs.
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Algebraic method for inverse
Algebraic method for inverse
Finding the inverse by solving for x in terms of y, swapping variables and simplifying.
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Graph method for inverse
Graph method for inverse
Finding the inverse by reflecting the function graph over y=x line.
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g(f(x)) = x
g(f(x)) = x
A verification step that g(x) is the inverse of f(x).
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f(x) is one-to-one
f(x) is one-to-one
Each input has a unique output and no two inputs share the same output.
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Domain of f⁻¹(x)
Domain of f⁻¹(x)
Same as the range of f(x).
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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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