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

Background

Topic: Functions and Relations

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 the relation is a function (each input has exactly one output).

Key Terms and Concepts:

• 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 input is paired with exactly one output.

Step-by-Step Guidance

1. List all the input (x) values and output (y) values from the given relation.

2. Identify the domain by collecting all unique x-values.

3. Identify the range by collecting all unique y-values.

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

5. Set up your answer for domain, range, and whether the relation is a function.

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Q2. Determine whether an equation represents a function.

Background

Topic: Functions from Equations

This question asks you to decide if a given equation defines y as a function of x, meaning for each x there is only one y.

Key Terms and Concepts:

• Function: Each input (x) has exactly one output (y).

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Step-by-Step Guidance

1. Look at the equation and try to solve for y in terms of x.

2. Check if, for every x-value, there is only one corresponding y-value.

3. If solving for y gives more than one value for some x, the equation does not represent a function.

4. Alternatively, consider the graph and apply the vertical line test.

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Q3. Evaluate functions.

Background

Topic: Function Evaluation

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

Key Terms and Formulas:

• Function notation: means the output of function f when input is x.

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

Step-by-Step Guidance

1. Write down the function expression, such as .

2. Identify the value to substitute for x (e.g., ).

3. Replace x in the function with the given value.

4. Simplify the expression to find the output.

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Q4. Obtain information about a function from its graph.

Background

Topic: Reading Graphs of Functions

This question tests your ability to interpret a function's graph to find key features such as intercepts, domain, range, and specific values.

Key Terms:

• x-intercept: Where the graph crosses the x-axis ().

• y-intercept: Where the graph crosses the y-axis ().

• Domain: All x-values covered by the graph.

• Range: All y-values covered by the graph.

Step-by-Step Guidance

1. Look at the graph and identify where it crosses the axes for intercepts.

2. Observe the leftmost and rightmost points to determine the domain.

3. Observe the lowest and highest points to determine the range.

4. Find any other requested values by reading from the graph.

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Q5. Identify intervals on which a function increases, decreases, or is constant.

Background

Topic: Increasing, Decreasing, and Constant Intervals

This question tests your ability to analyze a graph and determine where the function is going up, down, or staying the same as x increases.

Key Terms:

• Increasing: The function rises as x increases.

• Decreasing: The function falls as x increases.

• Constant: The function stays the same as x increases.

Step-by-Step Guidance

1. Examine the graph from left to right.

2. Identify intervals where the graph moves upward (increasing), downward (decreasing), or is flat (constant).

3. Write the intervals using interval notation, based on the x-values where the behavior changes.

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Q6. Use graphs to locate relative maxima or minima.

Background

Topic: Relative Extrema

This question asks you to find the highest or lowest points (relative maxima or minima) on a function's graph within a certain interval.

Key Terms:

• Relative Maximum: A point higher than all nearby points.

• Relative Minimum: A point lower than all nearby points.

Step-by-Step Guidance

1. Scan the graph for peaks (maxima) and valleys (minima).

2. Identify the x-values where these occur.

3. Record the coordinates of these points as relative maxima or minima.

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Q7. Identify even and odd functions from equations.

Background

Topic: Even and Odd Functions

This question tests your ability to determine if a function is even, odd, or neither by analyzing its equation.

Key Terms and Formulas:

• Even function: for all x in the domain.

• Odd function: for all x in the domain.

Step-by-Step Guidance

1. Substitute into the function: compute .

2. Compare to and .

3. If , the function is even. If , it is odd. Otherwise, it is neither.

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Q8. Perform transformations given the graph of a function.

Background

Topic: Transformations of Functions

This question tests your understanding of how shifting, stretching, or reflecting a function's graph changes its equation and appearance.

Key Terms and Formulas:

• Vertical shift:

• Horizontal shift:

• Reflection: or

• Stretch/Compression: or

Step-by-Step Guidance

1. Identify the type of transformation (shift, stretch, reflection) from the graph or equation.

2. Write the new function equation using the appropriate transformation formula.

3. Describe how the graph changes (e.g., "shifted up by 2 units").

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Q9. Perform translations on the standard quadratic function and square root function.

Background

Topic: Translations of Functions

This question focuses on shifting the graphs of and horizontally and/or vertically.

Key Formulas:

• Quadratic:

• Square root:

Step-by-Step Guidance

1. Identify the values of and for the translation.

2. Substitute and into the standard form to get the new function.

3. Describe the direction and magnitude of the translation.

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Q10. Perform transformations on the standard cubic function and cube root function.

Background

Topic: Transformations of Cubic and Cube Root Functions

This question tests your ability to apply shifts, stretches, and reflections to and .

Key Formulas:

• Cubic:

• Cube root:

Step-by-Step Guidance

1. Identify the transformation parameters (, , ).

2. Write the new function using these parameters.

3. Describe the effect of each parameter on the graph.

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Q11. Use transformations to graph rational functions.

Background

Topic: Graphing Rational Functions with Transformations

This question tests your ability to apply shifts, stretches, and reflections to rational functions like .

Key Formulas:

• Standard rational:

• Transformed:

Step-by-Step Guidance

1. Identify the values of , , and for the transformation.

2. Write the new function using these values.

3. Describe how the graph is shifted, stretched, or reflected.

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Q12. Find the domain of a function.

Background

Topic: Domain of Functions

This question tests your ability to determine all valid input values (x-values) for a given function, considering restrictions like division by zero or square roots of negative numbers.

Key Concepts:

• For rational functions, exclude x-values that make the denominator zero.

• For even roots, exclude x-values that make the radicand negative.

Step-by-Step Guidance

1. Identify any denominators and set them not equal to zero.

2. Identify any even roots and set the radicand greater than or equal to zero.

3. Solve these inequalities to find the domain.

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Q13. Combine functions using the algebra of functions, specifying domains.

Background

Topic: Algebra of Functions

This question tests your ability to add, subtract, multiply, or divide functions and to determine the domain of the resulting function.

Key Formulas:

• ,

Step-by-Step Guidance

1. Write the expressions for and .

2. Combine the functions as indicated (add, subtract, multiply, divide).

3. Determine the domain by considering the domains of and , and for division, where .

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Q14. Form composite functions.

Background

Topic: Composite Functions

This question tests your ability to form and evaluate composite functions, such as .

Key Formula:

Step-by-Step Guidance

1. Write down the expressions for and .

2. Substitute into wherever you see .

3. Simplify the resulting expression as much as possible.

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Q15. Determine domains for composite functions.

Background

Topic: Domain of Composite Functions

This question tests your ability to find the domain of a composite function, considering the domains of both and .

Key Concepts:

• The domain of is all x in the domain of such that is in the domain of .

Step-by-Step Guidance

1. Find the domain of .

2. Find the domain of .

3. Set in the domain of and solve for x.

4. The domain of the composite is the set of x-values that satisfy both conditions.

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Q16. Write functions as compositions.

Background

Topic: Decomposing Functions

This question asks you to express a given function as a composition of two or more simpler functions.

Key Concepts:

• Given , find and such that .

Step-by-Step Guidance

1. Look for an "inner" function inside .

2. Let be the inner part, and be the outer function applied to $g(x)$.

3. Check that reconstructs .

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Q17. Find the inverse of a function.

Background

Topic: Inverse Functions

This question tests your ability to find the inverse of a function, which "undoes" the action of the original function.

Key Steps:

• Replace with .

• Switch and .

• Solve for .

• Write for the inverse.

Step-by-Step Guidance

1. Write the function as .

2. Interchange and in the equation.

3. Solve the new equation for .

4. Express the result as .

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Q18. Use the horizontal line test to determine if a function has an inverse function.

Background

Topic: One-to-One Functions and Inverses

This question tests your ability to use the horizontal line test to determine if a function is one-to-one (and thus has an inverse that is also a function).

Key Concept:

• If every horizontal line crosses the graph at most once, the function is one-to-one and has an inverse function.

Step-by-Step Guidance

1. Visualize or draw the graph of the function.

2. Imagine drawing horizontal lines across the graph.

3. If any horizontal line crosses the graph more than once, the function does not have an inverse function.

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Q19. Use the graph of a one-to-one function to graph its inverse function.

Background

Topic: Graphs of Inverse Functions

This question tests your ability to graph the inverse of a function by reflecting the original graph over the line .

Key Concept:

• The graph of is the reflection of the graph of over the line .

Step-by-Step Guidance

1. Plot the graph of the original function .

2. Draw the line as a reference.

3. Reflect each point on to for .

4. Connect the reflected points to sketch the inverse graph.

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Q20. Find the inverse of a function and graph both functions on the same axes.

Background

Topic: Inverse Functions and Graphs

This question combines finding the inverse algebraically and graphing both the function and its inverse on the same set of axes.

Key Steps:

• Find the inverse function as in Q17.

• Graph both and on the same axes.

• Draw the line for reference.

Step-by-Step Guidance

1. Find the inverse function algebraically.

2. Plot several points for and .

3. Draw the line to show the reflection relationship.

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Q21. Solve applications involving linear functions.

Background

Topic: Applications of Linear Functions

This question tests your ability to model real-world situations with linear functions and solve for unknowns.

Key Concepts and Formulas:

• Linear function:

• Slope (): rate of change

• Y-intercept (): initial value

Step-by-Step Guidance

1. Identify the variables and what they represent in the context.

2. Write the linear function that models the situation.

3. Substitute given values to solve for unknowns.

4. Interpret the solution in the context of the problem.

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