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Questions and Answers
What is the general form of a quadratic function?
What is the general form of a quadratic function?
What is the operation to combine two functions f(x) and g(x) to get (f + g)(x)?
What is the operation to combine two functions f(x) and g(x) to get (f + g)(x)?
What is the name of the function that maps each input to a unique output?
What is the name of the function that maps each input to a unique output?
What is the point at which the graph of a function crosses the x-axis?
What is the point at which the graph of a function crosses the x-axis?
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What is the type of function that is represented as f(x) = a^x?
What is the type of function that is represented as f(x) = a^x?
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What is the range of a function?
What is the range of a function?
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What is the operation to combine two functions f(x) and g(x) to get (f ∘ g)(x)?
What is the operation to combine two functions f(x) and g(x) to get (f ∘ g)(x)?
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What is the name of the function that is both injective and surjective?
What is the name of the function that is both injective and surjective?
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What is the graph feature that a function approaches as x approaches a certain value?
What is the graph feature that a function approaches as x approaches a certain value?
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What is the type of function that is represented as f(x) = p(x) / q(x)?
What is the type of function that is represented as f(x) = p(x) / q(x)?
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Study Notes
Functions in Algebra
Definition
- A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
- In algebra, functions are often represented using variables and mathematical operations.
Types of Functions
- Linear Functions: of the form f(x) = mx + b, where m is the slope and b is the y-intercept.
- Quadratic Functions: of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
- Polynomial Functions: of the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where n is a non-negative integer.
- Rational Functions: of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions.
- Exponential Functions: of the form f(x) = a^x, where a is a constant.
- Logarithmic Functions: of the form f(x) = log_a(x), where a is a constant.
Function Operations
- Function Addition: (f + g)(x) = f(x) + g(x)
- Function Subtraction: (f - g)(x) = f(x) - g(x)
- Function Multiplication: (f * g)(x) = f(x) * g(x)
- Function Composition: (f ∘ g)(x) = f(g(x))
Function Properties
- Domain: the set of input values for which the function is defined.
- Range: the set of output values of the function.
- Injective (One-to-One): a function that maps each input to a unique output.
- Surjective (Onto): a function that maps each output to at least one input.
- Bijective: a function that is both injective and surjective.
Graphing Functions
- X-Intercept: the point at which the graph crosses the x-axis.
- Y-Intercept: the point at which the graph crosses the y-axis.
- Asymptotes: lines that the graph approaches as x approaches a certain value.
- Maxima and Minima: the highest and lowest points of the graph.
Functions in Algebra
Definition
- A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
- Functions are represented using variables and mathematical operations.
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.
- Polynomial Functions: f(x) = a_n x^n + a_(n-1) x^(n-1) +...+ a_1 x + a_0, where n is a non-negative integer.
- Rational Functions: f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions.
- Exponential Functions: f(x) = a^x, where a is a constant.
- Logarithmic Functions: f(x) = log_a(x), where a is a constant.
Function Operations
- Function Addition: (f + g)(x) = f(x) + g(x).
- Function Subtraction: (f - g)(x) = f(x) - g(x).
- Function Multiplication: (f * g)(x) = f(x) * g(x).
- Function Composition: (f ∘ g)(x) = f(g(x)).
Function Properties
- Domain: the set of input values for which the function is defined.
- Range: the set of output values of the function.
- Injective (One-to-One): a function that maps each input to a unique output.
- Surjective (Onto): a function that maps each output to at least one input.
- Bijective: a function that is both injective and surjective.
Graphing Functions
- X-Intercept: the point at which the graph crosses the x-axis.
- Y-Intercept: the point at which the graph crosses the y-axis.
- Asymptotes: lines that the graph approaches as x approaches a certain value.
- Maxima and Minima: the highest and lowest points of the graph.
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Description
This quiz covers the definition and types of functions in algebra, including linear, quadratic, and polynomial functions.
