Questions and Answers
What is a function?
A relation between a set of inputs (domain) and a set of possible outputs (range) where each input is related to exactly one output.
What is the notation typically used to denote a function?
f(x)
Which of the following is a type of function?
What is the general form of a linear function?
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What defines a rational function?
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The point where the graph crosses the x-axis is called the ______.
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The change in y over the change in x is called the ______.
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Even functions are symmetrical about the x-axis.
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What does the Vertical Line Test determine?
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What is the relationship between a function and its inverse?
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A function defined by multiple sub-functions is called a ______ function.
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What is the significance of the y-intercept?
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What is a logarithm?
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If $b^y = x$, then $\log_b(x) = ______$.
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Which of the following is the base for natural logarithms?
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What is the product property of logarithms?
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The graph of $y = \log_b(x)$ passes through the point $(1, 0)$.
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What is the change of base formula for logarithms?
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Which of the following is true about the range of logarithmic functions?
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To solve $\log_b(x) = y$, rewrite as $______ = b^y$.
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What is the base for common logarithms?
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Logarithmic functions grow quickly compared to polynomial and exponential functions.
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What does $\log_b(1)$ equal?
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What transformation does $y = \log_b(x - h) + k$ represent?
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Study Notes
Functions
- Definition: A function is a relation between a set of inputs (domain) and a set of possible outputs (range) where each input is related to exactly one output.
- Notation: Typically denoted as f(x), where f is the function name and x is the input value.
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Types of functions:
- Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
- Quadratic Functions: f(x) = ax² + bx + c, where a, b, and c are constants.
- Polynomial Functions: Functions of the form f(x) = a_nx^n + ... + a_1x + a_0.
- Rational Functions: Functions expressed as the ratio of two polynomials, f(x) = P(x)/Q(x).
- Exponential Functions: Functions of the form f(x) = a*b^x, where a is a constant and b > 0.
- Piecewise Functions: Functions defined by multiple sub-functions, each applied to a certain interval of the domain.
Graphs
- Coordinate Plane: Consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
- Plotting Points: Each point is represented as (x, y), where x is the horizontal position and y is the vertical position.
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Graphing Functions:
- Linear Graphs: Straight lines characterized by slope (m) and y-intercept (b).
- Quadratic Graphs: Parabolas that open upwards (a > 0) or downwards (a < 0).
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Transformation of Graphs:
- Shifts: Moving the graph left/right or up/down.
- Stretching/Compressing: Changing the steepness of the graph.
- Reflecting: Flipping the graph over an axis.
Important Concepts
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Domain and Range:
- Domain: All possible x-values for a function.
- Range: All possible y-values for a function.
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Intercepts:
- x-intercept: The point(s) where the graph crosses the x-axis (set y = 0).
- y-intercept: The point where the graph crosses the y-axis (set x = 0).
- Slope: Measures the steepness of a line, calculated as the rise over run (change in y/change in x).
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Symmetry:
- Even Functions: Symmetrical about the y-axis (f(x) = f(-x)).
- Odd Functions: Symmetrical about the origin (f(-x) = -f(x)).
Additional Notes
- Function Composition: Combining two functions where the output of one function becomes the input of another, denoted as (f∘g)(x) = f(g(x)).
- Inverse Functions: A function that reverses the effect of the original function, denoted as f⁻¹(x). A function and its inverse satisfy f(f⁻¹(x)) = x.
- Vertical Line Test: A method to determine if a graph represents a function; if any vertical line intersects the graph at more than one point, it is not a function.
Functions
- A function establishes a relationship between a set of inputs (domain) and outputs (range), with each input corresponding to one output.
- Notation is typically in the form f(x), where f represents the function and x is the input value.
- Types of functions include:
- Linear Functions: Represented as f(x) = mx + b, with m as slope and b as the y-intercept.
- Quadratic Functions: Expressed as f(x) = ax² + bx + c, involving constants a, b, and c.
- Polynomial Functions: General form f(x) = a_nx^n +...+ a_1x + a_0, where n is a non-negative integer.
- Rational Functions: Ratios of two polynomials, expressed as f(x) = P(x)/Q(x).
- Exponential Functions: Form f(x) = a*b^x, with a as a constant and b greater than 0.
- Piecewise Functions: Defined by multiple sub-functions applicable to specific intervals of the domain.
Graphs
- A coordinate plane consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis.
- Points on the graph are plotted as (x, y), where x indicates horizontal position and y indicates vertical position.
- Graphing types include:
- Linear Graphs: Straight lines deriving from slope (m) and y-intercept (b).
- Quadratic Graphs: Parabolas that may open upwards (a > 0) or downwards (a < 0).
- Transformations of graphs include:
- Shifts: Moving the graph left/right or up/down without altering shape.
- Stretching/Compressing: Adjusting the steepness of the graph.
- Reflecting: Flipping the graph over a specified axis.
Important Concepts
-
Domain and Range:
- Domain consists of all possible x-values for a function.
- Range includes all possible y-values for a function.
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Intercepts:
- x-intercept: Points at which the graph crosses the x-axis, found by setting y = 0.
- y-intercept: The point where the graph crosses the y-axis, determined by setting x = 0.
- Slope: Indicates the steepness of a line, calculated as the change in y (rise) over the change in x (run).
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Symmetry:
- Even Functions: Symmetrical about the y-axis, satisfying the condition f(x) = f(-x).
- Odd Functions: Symmetrical about the origin, fulfilling f(-x) = -f(x).
Additional Notes
- Function Composition: The process of combining two functions where the output of one serves as the input of the other, denoted (f∘g)(x) = f(g(x)).
- Inverse Functions: A function that undoes the effect of another, denoted f⁻¹(x), such that f(f⁻¹(x)) = x holds true.
- Vertical Line Test: A method for verifying if a graph represents a function; if any vertical line intersects the graph at more than once, it is not a function.
Logarithmic Functions Overview
- A logarithm is the exponent that a base must be raised to reach a specific number, expressed as ( b^y = x ) leads to ( \log_b(x) = y ).
- Common logarithmic bases include:
- Base 10: expressed as ( \log(x) )
- Base ( e ): known as the natural logarithm, expressed as ( \ln(x) )
- Base 2: represented as ( \log_2(x) )
Properties of Logarithms
- Product Property: The logarithm of a product is the sum of the logarithms, expressed as ( \log_b(MN) = \log_b(M) + \log_b(N) ).
- Quotient Property: The logarithm of a quotient is the difference of the logarithms, shown as ( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) ).
- Power Property: A logarithm of a power is the exponent times the logarithm of the base, given by ( \log_b(M^p) = p \cdot \log_b(M) ).
- Change of Base Formula: Allows conversion between bases, stated as ( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} ) for any base ( k ).
Graph Characteristics
- The graph for ( y = \log_b(x) ) passes through the point ( (1, 0) ), indicating that the logarithm of 1 is zero.
- A vertical asymptote exists at ( x = 0 ) which demonstrates the function's behavior as it approaches zero.
- Domain is limited to values where ( x > 0 ) while the range includes all real numbers.
Important Observations
- Logarithmic functions act as inverses of exponential functions; hence ( y = \log_b(x) ) can be rewritten as ( x = b^y ).
- Growth rate of logarithmic functions is significantly slower when compared to polynomial and exponential functions.
Applications of Logarithms
- Essential in solving equations involving exponentials.
- Practical applications include calculating pH levels in chemistry, measuring sound intensity in decibels, and determining the magnitude of earthquakes on the Richter scale.
Key Equations
- To solve ( \log_b(x) = y ), rewrite it as ( x = b^y ).
- For the equation ( b^y = x ), apply the logarithm to reformulate as ( y = \log_b(x) ).
Logarithmic Identities
- Key identities include:
- ( \log_b(1) = 0 ) indicating the result of a logarithm of one.
- ( \log_b(b) = 1 ) demonstrating the base of the logarithm.
- ( \log_b(b^k) = k ), confirming the relationship between a base raised to an exponent and its logarithm.
Transformations of Logarithmic Functions
- The transformation ( y = \log_b(x - h) + k ) results in a horizontal shift to the right by ( h ) and a vertical shift upward by ( k ).
Solving Logarithmic Equations
- To solve a logarithmic equation, isolate the logarithm and then exponentiate both sides to eliminate the logarithmic term.
- Utilize the properties of logarithms for simplification before solving for the variable.
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Description
This quiz covers the essential concepts of functions and their classification, including linear, quadratic, polynomial, rational, exponential, and piecewise functions. It also touches on the basics of graphing these functions on a coordinate plane.
