Q1. Find the domain and range of the relation, and determine whether the relation represents a function.

Background

Topic: Relations, Functions, Domain, and Range

This question tests your understanding of how to identify the domain (all possible input values) and range (all possible output values) of a relation, and how to determine if a relation is a function (each input has exactly one output).

Key Terms:

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Function: A relation where each element in the domain corresponds to exactly one element in the range.

Step-by-Step Guidance

1. List all the x-values (inputs) from the given relation. These make up the domain.

2. List all the y-values (outputs) from the relation. These make up the range.

3. Check if any x-value is paired with more than one y-value. If so, the relation is not a function.

4. Review the answer choices and match your findings to the correct statement about whether the relation is a function.

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Q2. Evaluate the function at the indicated values.

Background

Topic: Function Evaluation

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

Key Terms and Formulas:

• Function evaluation: Substitute the given value(s) for the variable in the function.

Step-by-Step Guidance

1. For part (a), substitute into : .

2. Simplify the expression by multiplying and combining like terms.

3. For part (b), substitute into : .

4. Distribute the and simplify the expression.

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Q3. Use the vertical line test to determine if is a function of in the graph.

Background

Topic: Functions and Graphs

This question tests your understanding of the vertical line test, which is used to determine if a graph represents a function.

Key Terms:

• Vertical Line Test: If any vertical line crosses the graph more than once, the graph does not represent a function.

Step-by-Step Guidance

1. Visualize or draw vertical lines at various -values on the graph.

2. Check if any vertical line intersects the graph at more than one point.

3. If all vertical lines intersect the graph at most once, is a function of .

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Q4. Classify the function as a polynomial, rational, or root function, and find its domain.

Background

Topic: Types of Functions and Domain

This question tests your ability to recognize the type of function and determine its domain.

Key Terms:

• Polynomial function: A function that is a sum of terms of the form .

• Rational function: A function that is the ratio of two polynomials.

• Root function: A function involving roots, such as square roots.

• Domain: The set of all real numbers for which the function is defined.

Step-by-Step Guidance

1. Examine the function's form to determine if it fits the definition of a polynomial, rational, or root function.

2. Check for any restrictions on the domain (such as division by zero or even roots of negative numbers).

3. Write the domain in interval notation.

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Q5. Find the x-intercept(s) and y-intercept of the function .

Background

Topic: Intercepts of Functions

This question tests your ability to find where a function crosses the x-axis (x-intercept) and y-axis (y-intercept).

Key Terms and Formulas:

• x-intercept: Set and solve for .

• y-intercept: Evaluate .

Step-by-Step Guidance

1. Set and solve for to find the x-intercept(s).

2. Substitute into to find the y-intercept.

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Q6. Use the graph to determine the domain and range of the function.

Background

Topic: Domain and Range from Graphs

This question tests your ability to read the domain and range of a function from its graph.

Key Terms:

• Domain: The set of all -values covered by the graph.

• Range: The set of all -values covered by the graph.

Step-by-Step Guidance

1. Look at the leftmost and rightmost points on the graph to determine the domain.

2. Look at the lowest and highest points on the graph to determine the range.

3. Express both answers in interval notation.

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Q7. Determine the intervals where the function is increasing, decreasing, or constant.

Background

Topic: Increasing, Decreasing, and Constant Intervals

This question tests your ability to analyze a graph and identify where the function rises, falls, or stays the same.

Key Terms:

• Increasing: The function rises as increases.

• Decreasing: The function falls as increases.

• Constant: The function remains at the same value as increases.

Step-by-Step Guidance

1. Examine the graph and identify intervals where the function moves upward (increasing), downward (decreasing), or is flat (constant).

2. Write each interval in interval notation.

3. Be careful to use open or closed intervals as appropriate based on the graph.

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Q8. Use the graph to find relative minimum and maximum values and their locations.

Background

Topic: Relative Extrema (Minimums and Maximums)

This question tests your ability to identify relative minimum and maximum points on a graph.

Key Terms:

• Relative minimum: A point where the function value is lower than all nearby points.

• Relative maximum: A point where the function value is higher than all nearby points.

Step-by-Step Guidance

1. Look for the lowest and highest points on the graph within a local region (not necessarily the absolute lowest/highest overall).

2. Record the -values where these points occur.

3. Record the corresponding -values for these points.

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Q9. Decide whether is even, odd, or neither.

Background

Topic: Even and Odd Functions

This question tests your ability to determine the symmetry of a function.

Key Terms and Formulas:

• Even function: for all in the domain.

• Odd function: for all in the domain.

Step-by-Step Guidance

1. Compute by substituting into the function.

2. Simplify and compare it to and .

3. Decide if the function is even, odd, or neither based on the comparison.

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Q10. Sketch the graph of and identify all properties that apply.

Background

Topic: Graphs of Basic Functions

This question tests your knowledge of the graph of the identity function and its properties.

Key Terms:

• Linear function: A function of the form .

• Odd function: .

• Domain and range: All real numbers for .

Step-by-Step Guidance

1. Recall that the graph of is a straight line through the origin with a slope of 1.

2. Identify the domain and range (all real numbers).

3. Check for symmetry (odd function) and whether the function is increasing everywhere.

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