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Questions and Answers
What is the domain of the function f(x) = √(x - 2)?
Which of the following is a quadratic function?
What is the composition of the functions f(x) = 2x + 1 and g(x) = x^2, denoted as (f ∘ g)(x)?
If f(x) = x^3 - 2x, is f(x) an even function, an odd function, or neither?
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If f(x) = 3x - 2 and g(x) = x + 1, what is the sum of the functions (f + g)(x)?
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Study Notes
Functions
Definition
- A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
- A function is a way of describing a relationship between variables.
Notation
- Functions are often denoted by letters such as f, g, or h.
- The function notation is f(x) = output, where x is the input.
Domain and Range
- The domain is the set of all possible inputs (x-values) of a function.
- The range is the set of all possible outputs (y-values) of a function.
Types of Functions
- Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
- Quadratic Functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
- Exponential Functions: f(x) = a^x, where a is a constant.
- Polynomial Functions: f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a positive integer.
Function Operations
- Addition: (f + g)(x) = f(x) + g(x)
- Subtraction: (f - g)(x) = f(x) - g(x)
- Multiplication: (f × g)(x) = f(x) × g(x)
- Composition: (f ∘ g)(x) = f(g(x))
Graphing Functions
- The graph of a function is a visual representation of the relationship between the input and output.
- The x-axis represents the domain, and the y-axis represents the range.
- The graph can be used to identify key features such as the x-intercept, y-intercept, and vertex.
Function Properties
-
Even and Odd Functions:
- Even function: f(-x) = f(x)
- Odd function: f(-x) = -f(x)
-
Increasing and Decreasing Functions:
- Increasing function: f(x) ≤ f(y) if x ≤ y
- Decreasing function: f(x) ≥ f(y) if x ≤ y
Functions
Definition
- A function is a relation between a set of inputs (domain) and a set of possible outputs (range), describing a relationship between variables.
Notation and Basics
- Functions are denoted by letters such as f, g, or h.
- Function notation is f(x) = output, where x is the input.
- Domain is the set of all possible inputs (x-values) of a function.
- Range is the set of all possible outputs (y-values) of a function.
Types of Functions
Linear Functions
- f(x) = mx + b, where m is the slope and b is the y-intercept.
Quadratic Functions
- f(x) = ax^2 + bx + c, where a, b, and c are constants.
Exponential Functions
- f(x) = a^x, where a is a constant.
Polynomial Functions
- f(x) = a_nx^n + a_(n-1)x^(n-1) +...+ a_1x + a_0, where n is a positive integer.
Function Operations
- (f + g)(x) = f(x) + g(x) (Addition)
- (f - g)(x) = f(x) - g(x) (Subtraction)
- (f × g)(x) = f(x) × g(x) (Multiplication)
- (f ∘ g)(x) = f(g(x)) (Composition)
Graphing Functions
- Graph of a function is a visual representation of the relationship between input and output.
- x-axis represents the domain, and y-axis represents the range.
- Graph can be used to identify key features such as x-intercept, y-intercept, and vertex.
Function Properties
Even and Odd Functions
- Even function: f(-x) = f(x)
- Odd function: f(-x) = -f(x)
Increasing and Decreasing Functions
- Increasing function: f(x) ≤ f(y) if x ≤ y
- Decreasing function: f(x) ≥ f(y) if x ≤ y
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Description
Understanding functions, notation, domain and range in mathematics. Learn about the definition, notation and important concepts related to functions.