Mathematics Concepts: Functions, Domain and Range

Definition of Domain and Range:
- The domain of a function is the complete set of possible values for the independent variable (typically $x$ ).
- The range of a function is the complete set of possible values for the dependent variable (typically $y$ ).
Function Situations:
- Given a description of a function in a certain context, I can identify and determine reasonable values for both the domain and the range relevant to that particular situation.

Further Exploration of Domain and Range:
- Continuing from previous lessons, I can further analyze situations where a function is defined and deduce its reasonable domain and range from those descriptions.

Understanding Piecewise Functions:
- I can interpret and understand the graph of a piecewise function, relating it to real-world situations by identifying various segments and their corresponding rules.
Graphing Piecewise Functions:
- When given specific rules defining sections of a piecewise function, I can sketch the graph of that function step by step.
Function Notation in Piecewise Functions:
- I can comprehend the notation used in piecewise functions, explaining how each piece corresponds to its specific condition or rule in the represented situation.
Characteristics of Piecewise Functions:
- I understand the criteria that categorize a function as a piecewise function, specifically how it consists of multiple distinct rules defined across different segments of its domain.

Working with Absolute Errors:
- Given a series of numerical guesses and a designated target number, I can calculate the absolute error, defined mathematically as the absolute value of the difference between the guessed number and the target number:
  $| ext{guess} - ext{target} |$ .
Creating Scatter Plots:
- I can create a scatter plot to visually represent absolute error data, displaying how different guesses relate to their corresponding absolute errors.
Analyzing Scatter Plots:
- I can examine and describe features within the scatter plot that provide insights into the nature of the absolute errors, such as trends or patterns.
General Relationships in Absolute Values:
- I can articulate the general relationship between guesses and their absolute errors using both verbal descriptions and mathematical equations to capture the essence of the interaction.

Impact of Adding a Number:
- I can describe how adding a constant number to the expression that defines an absolute value function affects the graph and the values it takes.
Understanding Distance through Absolute Value:
- I can explain how the concept of an absolute value function can be interpreted in terms of distance from a certain point on the real number line, reinforcing its practical implications.
Interpreting Absolute Value Functions:
- When presented with an absolute value function expressed in written or function notation, I can decipher its meaning, create a corresponding table of values, and produce a graph that accurately represents the behavior of the function.

Concept of Inverse Functions:
- I have a solid understanding of what constitutes an inverse function, which effectively reverses the mapping of the original function. For a function $f(x)$ , its inverse is typically denoted $f^{-1}(x)$ , where the relation $f(f^{-1}(x)) = x$ holds true for every $x$ in the domain of $f$ .
Finding Inverse Functions:
- I can identify methods to determine the inverse function of a provided function through algebraic manipulation, substitution, and graphical understanding.
Describing Inverse Functions in Context:
- When given a linear function that describes a specific situation, I can articulate the inverse function through word problems and equations, emphasizing its practical application in the scenario discussed.

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