Function Flashcards
(36 cards)
What is the definition of the signum function?
A function from R to R whose value at any x in R is 1 if x is positive, 0 if x is zero and -1 if x is negative, denoted by sgn(x)
The signum function is used in various mathematical contexts, including analysis and signal processing.
How is the Heaviside function defined?
A function from R to R whose value at any non-negative number x is 1 and whose value at all other real numbers x is 0, denoted by H(x)
The Heaviside function is often used in control theory and signal processing.
What is the definition of the modulus or absolute value function?
The modulus or absolute value function on R is denoted by mod(x)
The absolute value function returns the non-negative value of a number.
What is the general form of a linear function?
A linear function is of the form f(x) = ax + b, where a and b are constants
Linear functions represent straight lines in a Cartesian plane.
What is a polynomial function?
A polynomial function p(x) of degree n is defined as the sum of terms of the form a_ix^i, where a_i are constants
Polynomial functions can have varying degrees, influencing their shape and behavior.
Define a rational function.
A rational function is a function of the form p(x)/q(x), where p(x) and q(x) are polynomial functions
Rational functions can exhibit asymptotic behavior and discontinuities.
What are trigonometric functions?
Functions including sin(x), cos(x), tan(x) and their derived functions such as cosec(x), sec(x), cot(x)
Trigonometric functions are fundamental in geometry and periodic phenomena.
List the hyperbolic functions.
- sinh(x)
- cosh(x)
- tanh(x)
- coth(x)
- sech(x)
- csch(x)
Hyperbolic functions are analogs of trigonometric functions for the hyperbola.
What is the form of an exponential function?
An exponential function is of the form a^x, where a is a constant
Exponential functions are characterized by rapid growth or decay.
What is the form of a logarithmic function?
A logarithmic function is of the form log_a(x), where a is the base
Logarithmic functions are the inverses of exponential functions.
What is the definition of the domain of a function?
The set of values of x at which a function is defined
The domain is crucial for understanding the behavior and limits of functions.
What is the definition of the range of a function?
The corresponding set of values that the function can produce
The range indicates the output values for given inputs within the domain.
What is a single-valued function?
A set of ordered pairs in which no two different pairs have the same first entry
Single-valued functions ensure that each input corresponds to exactly one output.
Define a constant function.
A function of the form f(x) = k, where k is a constant
The graph of a constant function is a horizontal line.
What is an identity function?
A function of the form f(x) = x, indicating that the output is the same as the input
The identity function serves as a fundamental concept in function theory.
What is function composition?
The evaluation of one function in terms of another, denoted by (f ∘ g)(x) = f(g(x))
Composition allows for the combining of functions to create new functions.
What is the process to find the inverse of a function?
- Convert the function to an equation by replacing f(x) with y.
- Make x the subject.
- Interchange x and y.
The inverse of a function undoes the action of the original function.
What is the understood domain of a function?
The largest subset of real numbers in which the function is defined
Understanding the domain is essential for analyzing function behavior.
What is the definition of the signum function?
A function from R to R whose value at any x in R is 1 if x is positive, 0 if x is zero and -1 if x is negative, denoted by sgn(x)
The signum function is used in various mathematical contexts, including analysis and signal processing.
How is the Heaviside function defined?
A function from R to R whose value at any non-negative number x is 1 and whose value at all other real numbers x is 0, denoted by H(x)
The Heaviside function is often used in control theory and signal processing.
What is the definition of the modulus or absolute value function?
The modulus or absolute value function on R is denoted by mod(x)
The absolute value function returns the non-negative value of a number.
What is the general form of a linear function?
A linear function is of the form f(x) = ax + b, where a and b are constants
Linear functions represent straight lines in a Cartesian plane.
What is a polynomial function?
A polynomial function p(x) of degree n is defined as the sum of terms of the form a_ix^i, where a_i are constants
Polynomial functions can have varying degrees, influencing their shape and behavior.
Define a rational function.
A rational function is a function of the form p(x)/q(x), where p(x) and q(x) are polynomial functions
Rational functions can exhibit asymptotic behavior and discontinuities.
