Questions and Answers
What is the introduction to functions?
lesson 1
Is the relation in the mapping diagram a function? Why or why not?
No, there is an input that maps to more than one output.
Is the relation that maps animals to foods a function? Why or why not?
No, there is an input that has more than one output.
Which of the following sets of ordered pairs does not define a function?
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Which two solutions to the equation $x^2 + y^2 = 20$ can be used to show that it is not a function?
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Does the table represent y as a function of x? Why or why not?
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Which reasons prove that the graph represents a function?
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Does the graph represent a function? Why or why not?
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Which two ordered pairs generated by the equation $x = y^2 - 1$ can be used to show it is not a function?
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What is the set of input values from the relation mapping x-values to y-values: x: 0, 2, 4, 6, y: 0, 1, 2, 3?
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Which of the following relations is also a function?
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What are the domain and range of the function represented by the following graph?
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What is the range of $f(x)$ if $f(x) = x^2$?
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Which function rule defines $f(x)$?
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How can you evaluate $ff(15)$ for the function $f(x) = 2(x + 3)$?
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What is the common difference between the terms in the sequence {17, 11, 5, −1, −7...}?
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Study Notes
Functions Overview
- Functions map each input to exactly one output.
- A relation is a function if no input maps to multiple outputs.
Identifying Functions
- Mapping Diagrams & Ordered Pairs: To determine if relations are functions, check for unique outputs for each input.
- Relation examples that do not define functions include instances where an input has multiple outputs.
Function Notation and Properties
- Domain: Set of all possible input values.
- Range: Set of all possible output values.
- Function notation examples include f(x) and g(n), representing mathematical expressions.
Graph Characteristics
- Vertical Line Test: A graph represents a function if a vertical line intersects the graph at most once.
- Functions can be identified through graphical representation, ensuring no repeating x-values yield different y-values.
Sequences and Recursive Formulas
- Sequences can be represented explicitly or recursively.
- Common difference in arithmetic sequences is the consistent value added/subtracted between terms.
Evaluating Functions
- To evaluate a function f at a certain value, substitute that value into the function's expression.
- Function outputs are determined through proper substitution and simplification.
Function Rules and Situations
- Various scenarios (e.g., earnings, distances) can be modeled using functions.
- Arithmetic sequences can be represented through both explicit formulas (an = a1 + (n - 1)d) and recursive formulas (an = an-1 + d).
Explicit vs. Recursive Definitions
- Explicit formulas provide a direct way to compute the nth term of a sequence, while recursive formulas rely on previous terms.
- Transitioning between explicit and recursive formulas is fundamental in understanding sequences.
Special Functions
- Quadratic and linear functions have distinct characteristics in their graphs and equations.
- Specific functions like f(x) = x^2 demonstrate unique properties, such as a range of non-negative values.
Practical Applications
- Real-world scenarios often utilize functions, such as calculating pay based on hours worked or predicting growth patterns.
- Each situation provides context for how functions are structured and used.
Summary
- Understanding functions requires a grasp of their definitions, properties, and the ability to differentiate between valid functions and non-functions based on graphical, numerical, and contextual clues.
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Description
This quiz covers the fundamentals of functions, including their definitions, properties, and methods for identifying them through mapping diagrams and graphs. You will also learn about the domain and range, as well as sequences and their representations.
