Q1. Give the domain and range of the relation {(-1, -1), (7, 4), (3, 0), (-5, -1)}.

Background

Topic: Relations, Domain, and Range

This question tests your understanding of how to identify the domain (all possible x-values) and range (all possible y-values) from a set of ordered pairs.

Key Terms:

• Domain (D): The set of all first elements (x-values) in the relation.

• Range (R): The set of all second elements (y-values) in the relation.

Step-by-Step Guidance

1. List all the x-values from each ordered pair to determine the domain.

2. List all the y-values from each ordered pair to determine the range.

3. Write the domain and range as sets, making sure not to repeat any values.

4. Check your sets to ensure all values are included and there are no duplicates.

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Q2. Give the domain and range of each relation in set notation (for a given set of ordered pairs).

Background

Topic: Set Notation for Domain and Range

This question asks you to express the domain and range of a relation using set notation, which is a standard way to list elements in mathematics.

Key Terms:

• Set Notation: Curly braces are used to list elements of a set.

Step-by-Step Guidance

1. Identify all unique x-values from the relation to form the domain set.

2. Identify all unique y-values from the relation to form the range set.

3. Write each set using curly braces, separating elements with commas.

4. Double-check for any repeated values and remove duplicates.

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Q3. Give the domain and range of each relation in interval notation.

Background

Topic: Interval Notation for Domain and Range

This question tests your ability to express the domain and range as intervals, which is especially useful for continuous data or when values form a sequence.

Key Terms:

• Interval Notation: Uses parentheses ( ) for open intervals and brackets [ ] for closed intervals to describe all numbers between endpoints.

Step-by-Step Guidance

1. List all x-values (for domain) and y-values (for range) from the relation.

2. Identify the smallest and largest values for each set.

3. Determine if the endpoints should be included (use [ ] for included, ( ) for not included).

4. Write the interval notation for both domain and range.

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Q4. Determine whether the given relation is a function (using a table or set of ordered pairs).

Background

Topic: Functions and the Definition of a Function

This question tests your understanding of what makes a relation a function: each input (x-value) must correspond to exactly one output (y-value).

Key Terms:

• Function: A relation where every x-value is paired with only one y-value.

• Vertical Line Test: A graphical method to determine if a relation is a function.

Step-by-Step Guidance

1. Examine each x-value in the relation.

2. Check if any x-value is paired with more than one y-value.

3. If all x-values are unique or each x-value maps to only one y-value, the relation is a function.

4. If any x-value maps to more than one y-value, the relation is not a function.

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Q5. Write each equation explicitly in terms of x. Then, indicate whether the equation is a function.

Background

Topic: Explicit and Implicit Equations; Functions

This question tests your ability to rewrite equations so that y is expressed explicitly in terms of x, and to determine if the resulting equation represents a function.

Key Terms and Formulas:

• Explicit Equation: An equation where y is written as a function of x (e.g., ).

• Implicit Equation: An equation where x and y are mixed together (e.g., ).

• Function: Each x-value must correspond to only one y-value.

Step-by-Step Guidance

1. Isolate y on one side of the equation to write it explicitly in terms of x.

2. Solve for y, using algebraic operations as needed (addition, subtraction, multiplication, division, square roots, etc.).

3. Examine the resulting expression: does each x-value produce only one y-value?

4. If so, the equation is a function; if not (e.g., ), it is not a function.

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Q6. Evaluate each function for the given value (e.g., f(x) = x^2 - 6x + 23; f(-5)).

Background

Topic: Function Notation and Evaluation

This question tests your ability to substitute a given value into a function and simplify the result.

Key Terms and Formulas:

• Function Notation: means the value of the function f at x.

• Substitution: Replace x with the given value in the function expression.

Step-by-Step Guidance

1. Write the function expression and identify the value to substitute for x.

2. Replace every x in the function with the given value.

3. Carefully perform the arithmetic operations (exponents, multiplication, addition, etc.).

4. Simplify the expression step by step, but stop before the final calculation.

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Q7. Using the graph to the left, find each function value (e.g., f(1), f(-2), f(5), f(0)).

Background

Topic: Reading Function Values from a Graph

This question tests your ability to interpret a graph and find the output value (y) for a given input (x).

Key Terms:

• Function Value: The y-coordinate corresponding to a given x-coordinate on the graph.

Step-by-Step Guidance

1. Locate the given x-value on the horizontal axis of the graph.

2. Move vertically from this x-value until you reach the graph of the function.

3. Read the corresponding y-value at this point; this is the function value.

4. Repeat for each requested x-value.

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