Q1. Determine if the relation is a function. Give the domain and range.

Background

Topic: Relations and Functions

This question tests your understanding of what makes a relation a function, and how to identify the domain and range from a set of ordered pairs.

Key Terms:

• Relation: A set of ordered pairs.

• Function: A relation where each input (x-value) corresponds to exactly one output (y-value).

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

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

Step-by-Step Guidance

1. List all the x-values from the given ordered pairs: .

2. Check if any x-value is repeated. If each x-value is unique, the relation is a function.

3. List all the y-values from the ordered pairs to find the range.

4. Write the domain and range as sets, using curly braces .

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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 Formula:

Step-by-Step Guidance

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

2. Multiply and then subtract $8$.

3. For part (b), substitute : .

4. Multiply and then subtract $8$.

5. For part (c), substitute into the function: .

6. Expand and then subtract $8$.

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

Background

Topic: Functions and Graphs

This question tests your ability to use the vertical line test to determine if a graph represents a function.

Key Concept:

• 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 x-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, y is a function of x.

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Q4. Classify the function and find its domain.

Background

Topic: Types of Functions and Domain

This question tests your ability to classify functions and determine their domain.

Key Terms:

• Polynomial Function: A function made up of terms with non-negative integer exponents.

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

• Interval Notation: A way to express the domain using intervals.

Step-by-Step Guidance

1. Identify the type of function by looking at the exponents and terms.

2. Check for any restrictions on x (such as division by zero or square 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 .

Background

Topic: Intercepts of Rational Functions

This question tests your ability to find where a function crosses the axes.

Key Formulas:

• x-intercept: Set and solve for x.

• y-intercept: Set and solve for .

Step-by-Step Guidance

1. For x-intercept(s), set : .

2. Solve for x where the numerator is zero, but denominator is not zero.

3. For y-intercept, substitute into : .

4. Simplify the expression for 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 domain and range from a graph.

Key Terms:

• Domain: All x-values for which the function is defined.

• Range: All y-values that the function takes.

• Interval Notation: Used to express domain and range.

Step-by-Step Guidance

1. Look at the graph and identify the leftmost and rightmost x-values for the domain.

2. Identify the lowest and highest y-values for the range.

3. Write the domain and range 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 describe where the function changes behavior.

Key Terms:

• Increasing: Function values go up as x increases.

• Decreasing: Function values go down as x increases.

• Constant: Function values stay the same as x increases.

Step-by-Step Guidance

1. Examine the graph and identify intervals where the function rises, falls, or stays flat.

2. Write each interval in interval notation.

3. Check for any overlapping or missing intervals.

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Q8. Find the relative minimum and maximum values and their locations.

Background

Topic: Relative Extrema

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

Key Terms:

• Relative Minimum: The lowest point in a local region of the graph.

• Relative Maximum: The highest point in a local region of the graph.

Step-by-Step Guidance

1. Look for points on the graph where the function changes from decreasing to increasing (minimum) or increasing to decreasing (maximum).

2. Identify the x-values where these changes occur.

3. Find the corresponding y-values for these x-values.

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Q9. Determine whether the function is even, odd, or neither.

Background

Topic: Even and Odd Functions

This question tests your ability to classify functions based on symmetry.

Key Concepts:

• Even Function: for all x; symmetric about the y-axis.

• Odd Function: for all x; symmetric about the origin.

Step-by-Step Guidance

1. Check the function or graph for y-axis symmetry (even) or origin symmetry (odd).

2. Test and compare to and .

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Q10. Sketch the graph of and identify its properties.

Background

Topic: Graphs of Rational Functions

This question tests your ability to recognize the graph and properties of the reciprocal function.

Key Properties:

• Domain: All real numbers except .

• Range: All real numbers except .

• Odd Function: Symmetric about the origin.

Step-by-Step Guidance

1. Sketch the graph with two branches, one in each quadrant.

2. Identify vertical and horizontal asymptotes.

3. List the domain and range, and check for symmetry.

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Q11. Sketch the graph of for .

Background

Topic: Graphs with Restricted Domains

This question tests your ability to graph a function with a domain restriction.

Key Concepts:

• Restricted Domain: Only graph the function for .

Step-by-Step Guidance

1. Draw the parabola but only for values where .

2. Indicate the endpoint at (open or closed circle as appropriate).

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Q12. Write the domain and define the function using an inequality for a graph with domain .

Background

Topic: Domain and Function Definition

This question tests your ability to express domain in interval notation and as an inequality.

Key Concepts:

• Interval Notation:

• Inequality:

Step-by-Step Guidance

1. Write the domain in interval notation.

2. Express the domain as an inequality.

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Q13. Write a function for a graph shifted horizontally and vertically from .

Background

Topic: Transformations of Functions

This question tests your ability to write equations for shifted graphs.

Key Formula:

Step-by-Step Guidance

1. Identify the horizontal shift and vertical shift from the graph.

2. Write the function in the form .

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Q14. Sketch the graph of and determine how points move.

Background

Topic: Function Reflections

This question tests your understanding of reflecting a graph across the y-axis.

Key Concept:

• Reflection: reflects the graph across the y-axis.

Step-by-Step Guidance

1. For each point on the original graph, the reflected point is .

2. Apply this to each given point to find its new location.

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Q15. Sketch the graph of and describe the transformation.

Background

Topic: Exponential Functions and Transformations

This question tests your ability to recognize vertical stretches and compressions.

Key Formula:

Step-by-Step Guidance

1. Compare to the basic function to determine the stretch/compression factor.

2. Sketch the graph with the appropriate vertical transformation.

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Q16. Evaluate given , .

Background

Topic: Function Operations

This question tests your ability to multiply two functions and evaluate at a specific value.

Key Formula:

Step-by-Step Guidance

1. Find by substituting into .

2. Find by substituting into .

3. Multiply the two results: .

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Q17. Use the graphs of and to evaluate expressions.

Background

Topic: Function Operations from Graphs

This question tests your ability to read values from graphs and perform operations.

Key Concepts:

• Read and values at specific x-values.

• Perform addition, subtraction, multiplication, or division as indicated.

Step-by-Step Guidance

1. Locate the x-value on the graph and find and .

2. Apply the operation (add, subtract, multiply, divide).

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Q18. For , , find , , , and .

Background

Topic: Function Operations

This question tests your ability to combine functions using addition, subtraction, multiplication, and division.

Key Formulas:

Step-by-Step Guidance

1. Write each operation using the given functions.

2. Simplify each expression as much as possible.

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Q19. Evaluate the composite function at , given .

Background

Topic: Composite Functions

This question tests your ability to evaluate a function composed with itself.

Key Formula:

Step-by-Step Guidance

1. First, find by substituting into .

2. Next, substitute the result into again: .

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Q20. Find for , .

Background

Topic: Composite Functions

This question tests your ability to compose two functions and simplify the result.

Key Formula:

Step-by-Step Guidance

1. Substitute into : .

2. Replace with and simplify.

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Q21. Determine whether the graph of the function is one-to-one.

Background

Topic: One-to-One Functions

This question tests your ability to recognize one-to-one functions from their graphs.

Key Concept:

• One-to-One Function: Each y-value corresponds to only one x-value; passes the horizontal line test.

Step-by-Step Guidance

1. Draw horizontal lines at various y-values on the graph.

2. If any horizontal line crosses the graph more than once, the function is not one-to-one.

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Q22. Determine whether and are inverse functions by evaluating and .

Background

Topic: Inverse Functions

This question tests your ability to check if two functions are inverses by composing them.

Key Formula:

If and are inverses, then and .

Step-by-Step Guidance

1. Write and .

2. Find by substituting into .

3. Find by substituting into .

4. Check if either composition simplifies to .

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