Questions and Answers
What does 'domain' refer to in functions?
Input (x-values)
What does 'range' refer to in functions?
Output (y-values)
What is a 'one-to-one' function?
Each input has exactly one unique output.
What does 'onto' mean regarding functions?
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What characterizes an 'onto/one-to-one' function?
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What does 'discrete' mean in mathematics?
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What does 'continuous' mean in terms of functions?
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What is the 'Vertical Line Test'?
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What is 'function notation'?
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What does it mean to 'evaluate' a function?
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What is the formula for a 'linear function'?
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What is a 'line of symmetry'?
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What is a 'turning point' on a graph?
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What is a 'relative minimum' on a graph?
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What is a 'relative maximum' on a graph?
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What are 'zeros' in relation to a graph?
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What is an 'x-intercept'?
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What is a 'y-intercept'?
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What does 'positive' signify in a graph?
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What does 'negative' signify in a graph?
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What does 'increasing' mean regarding a function?
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What does 'decreasing' mean regarding a function?
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What is 'end behavior'?
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What does 'continuity' refer to in functions?
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What are 'extrema' in calculus?
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What does 'symmetry' mean concerning graphs?
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Study Notes
Functions and Continuity Concepts
- Domain: Refers to the set of all possible input values (x-values) for a function.
- Range: Represents all possible output values (y-values) produced by a function.
Characteristics of Functions
- One-to-one: Each input has a unique output; no two different inputs share the same output.
- Onto: Every possible output is associated with at least one input from the domain.
- Onto/one-to-one: Each input has a unique output, and each output corresponds to exactly one input, ensuring a bijective relationship.
Types of Functions
- Discrete: Consists of a finite number of separate points that are not connected.
- Continuous: Contains an infinite number of points that are connected, extending indefinitely in both directions.
Graphical Analysis
- Vertical Line Test: A method to determine if a curve is a function; if any vertical line intersects the graph more than once, it is not a function.
- Function Notation: Defined by replacing y with f(x); a way to express functions clearly.
- Evaluate: The process of substituting a specific value or expression for x in a function, often referred to as "Plug and Chug."
Linear Function
- Linear Function Equation: Represented by f(x) = mx + b, where m is the slope and b is the y-intercept.
Graph Characteristics
- Line of Symmetry: A vertical line that acts as a folding point, dividing the graph into two mirror-image halves.
- Turning Point: The point at which a graph changes direction, indicating local extrema.
- Relative Minimum: A point where the graph transitions from decreasing to increasing, representing a local low.
- Relative Maximum: Occurs when the graph changes from increasing to decreasing, indicating a local high.
Intercepts and Axes
- Zeros: Points where the graph intersects the x-axis; equivalent to x-intercepts.
- X-Intercept: The specific point(s) where the graph crosses the x-axis.
- Y-Intercept: The point(s) where the graph intersects the y-axis.
Function Behavior
- Positive: Regions of the graph lying above the x-axis indicate positive function values.
- Negative: Regions below the x-axis reflect negative function values.
- Increasing: Describes the trend where, as input values (x) increase, the function values (f(x)) also increase.
- Decreasing: Indicates that function values decrease as the input values increase.
Long-Term Behavior
- End Behavior: Describes the behavior of the function values as x approaches positive or negative infinity.
- Continuity: Refers to functions that are connected with no breaks, resulting in smooth curves or lines.
- Extrema: The maximum and minimum values of a function, outlining the highest and lowest points on a graph.
- Symmetry: Refers to functions exhibiting a vertical line that divides the graph into two symmetrical halves.
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Explore essential definitions related to functions and continuity with these flashcards. Each card succinctly defines important concepts such as domain, range, and types of functions, facilitating a quick review for students. Ideal for anyone preparing for algebra exams or seeking to reinforce their understanding of mathematical functions.