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Questions and Answers
Given the functions $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, what is the result of $(g \circ f)(2)$?
Given the functions $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, what is the result of $(g \circ f)(2)$?
- 12
- 9
- 3
- 7 (correct)
Which of the following function types is characterized by having a potential vertical asymptote?
Which of the following function types is characterized by having a potential vertical asymptote?
- Rational functions (correct)
- Exponential functions
- Polynomial functions
- Trigonometric functions
If $f(x) = x^3$ and $g(x) = \sqrt{x}$, which of the following operations results in a function that is not defined for all real numbers?
If $f(x) = x^3$ and $g(x) = \sqrt{x}$, which of the following operations results in a function that is not defined for all real numbers?
- $(g / f)(x)$ (correct)
- $(f - g)(x)$
- $(f * g)(x)$
- $(f + g)(x)$
Which of the following relations is NOT a function, according to the vertical line test?
Which of the following relations is NOT a function, according to the vertical line test?
If $h(x)$ is a piecewise function where $h(x) = x + 1$ for $x < 0$, and $h(x) = x^2$ for $x \ge 0$, what is the value of $h(-1) + h(2)$?
If $h(x)$ is a piecewise function where $h(x) = x + 1$ for $x < 0$, and $h(x) = x^2$ for $x \ge 0$, what is the value of $h(-1) + h(2)$?
Which statement accurately describes the relationship between a function's range and its codomain?
Which statement accurately describes the relationship between a function's range and its codomain?
A function is considered bijective if it is:
A function is considered bijective if it is:
Given a function f: A → B, what condition must be met for f to be a valid function?
Given a function f: A → B, what condition must be met for f to be a valid function?
What is the defining characteristic of a constant function?
What is the defining characteristic of a constant function?
If a function f is bijective, what can be inferred about its inverse function f⁻¹?
If a function f is bijective, what can be inferred about its inverse function f⁻¹?
Consider two functions f and g. What is involved in 'composition of functions'?
Consider two functions f and g. What is involved in 'composition of functions'?
Which of the following equations represents a linear function?
Which of the following equations represents a linear function?
If a function has a range that is a subset of its codomain, and not all elements of its codomain are in the range, the given function is:
If a function has a range that is a subset of its codomain, and not all elements of its codomain are in the range, the given function is:
Flashcards
Polynomial function
Polynomial function
A function that can be expressed as a sum of terms, each of which is a constant multiplied by a non-negative integer power of the variable.
Rational function
Rational function
A function formed by dividing two polynomial functions.
Exponential function
Exponential function
A function where the input variable appears as an exponent.
Logarithmic function
Logarithmic function
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Vertical line test
Vertical line test
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What is a function?
What is a function?
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What is the domain?
What is the domain?
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What is the codomain?
What is the codomain?
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What is range?
What is range?
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What is a one-to-one function?
What is a one-to-one function?
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What is an onto function?
What is an onto function?
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What is a bijective function?
What is a bijective function?
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What is a constant function?
What is a constant function?
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Study Notes
Definition of a Function
- A function is a relation where each input (domain) is associated with exactly one output (codomain).
- A function from set A to set B maps every element 'a' in A to exactly one element 'b' in B.
- Functions are often written as f: A → B, where 'f' is the function name, 'A' is the domain, and 'B' is the codomain.
Function Notation
- f(x) = y represents the output 'y' for input 'x' of function 'f'.
- 'x' is the input or argument of the function.
- 'y' is the output or value of the function for input 'x'.
- An alternative notation is f(x) for the function value.
Types of Functions
- One-to-one (Injective): If f(x₁) = f(x₂), then x₁ = x₂. Each output is associated with at most one input.
- Onto (Surjective): Every element in the codomain is associated with at least one element in the domain. The function's range equals the codomain.
- Bijective: A function that is both one-to-one and onto. Every domain element maps to a unique codomain element, and every codomain element is mapped to by at least one domain element.
- Constant Functions: Produce the same output for all inputs.
- Linear Functions: Constant rate of change, graphed as a straight line. Form: f(x) = mx + b, where m is the slope and b is the y-intercept.
Properties of Functions
- Domain: The set of all possible input values (x-values).
- Range: The set of all possible output values (y-values) corresponding to inputs in the domain.
- Codomain: The set of all possible output values a function could map to. The range is a subset of the codomain.
- Inverse Functions: If 'f' is bijective, an inverse function (f⁻¹) exists, reversing the mapping. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
- Composition of Functions: Combining multiple functions, where the output of one is the input of the next. If f: A → B and g: B → C, (g o f)(x) = g(f(x)).
Special Types of Functions (Examples)
- Polynomial Functions: Functions written as polynomials, e.g., f(x) = x² + 2x + 1.
- Rational Functions: Functions that are quotients of polynomials, e.g., f(x) = (x+1)/(x-2).
- Exponential Functions: f(x) = ax, where 'a' is a constant.
- Logarithmic Functions: Inverses of exponential functions, e.g., f(x) = loga(x).
- Trigonometric Functions: Relate angles and sides of a right triangle (sin, cos, tan).
- Piecewise Functions: Defined by different expressions in different intervals of the domain.
Basic 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 Division: (f / g)(x) = f(x) / g(x), provided g(x) ≠ 0
Graphing Functions
- A visual representation of the function's input-output relationship.
- Plot points (x, f(x)) on a coordinate system.
- Graph features (intercepts, symmetry, maxima/minima, asymptotes) reflect function properties.
Determining if a Relation is a Function
- Vertical Line Test: If a vertical line intersects the graph more than once, the relation is not a function.
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