Q1. Complete the table if is an even function.

Background

Topic: Even and Odd Functions

This question tests your understanding of even functions and how their symmetry properties affect function values for positive and negative inputs.

Key Terms and Formulas:

• Even Function: A function is even if for all in the domain.

Step-by-Step Guidance

1. Identify the given -values and their corresponding values in the table.

2. Recall that for an even function, . This means the value at is the same as at .

3. Use the given values to fill in the missing entries by matching each with its counterpart.

4. For , remember that is always its own pair, so use the given value if available.

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Q2. Evaluate the piecewise function at :

Background

Topic: Piecewise Functions

This question tests your ability to evaluate a function defined by different expressions over different intervals of .

Key Terms and Formulas:

• Piecewise Function: A function defined by different expressions depending on the value of .

Step-by-Step Guidance

1. For each value of (, $0), determine which piece of the function applies by checking the interval conditions.

2. For , check which condition it satisfies and substitute into the corresponding expression.

3. For , check which condition it satisfies and substitute into the corresponding expression.

4. For , check which condition it satisfies and substitute into the corresponding expression.

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Q3. Simplify the following expressions:

Background

Topic: Exponents and Radicals

This question tests your ability to simplify expressions involving exponents, including fractional and negative exponents.

Key Terms and Formulas:

• Exponent Rules:

Step-by-Step Guidance

1. For each expression, identify which exponent rules apply (e.g., power of a power, product of powers, negative exponents).

2. Rewrite each expression using the appropriate rules, simplifying step by step.

3. For fractional exponents, remember that means the th root of raised to the th power.

4. Stop before the final numeric simplification; set up the expression for the last calculation.

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Q4. Write a piecewise function for the given graph.

Background

Topic: Piecewise Functions from Graphs

This question tests your ability to interpret a graph and write a piecewise function that matches its behavior.

Key Terms and Formulas:

• Piecewise Function: A function defined by different expressions over different intervals of .

• Look for changes in the graph's behavior (e.g., open/closed circles, slope changes) to determine interval boundaries.

Step-by-Step Guidance

1. Identify the intervals on the -axis where the graph changes its rule (e.g., at , , ).

2. For each interval, determine the equation of the segment (e.g., linear, constant, quadratic) by analyzing the graph's shape and points.

3. Write the function rule for each interval, using the correct domain restrictions.

4. Stop before writing the complete piecewise function; set up the intervals and describe the process for finding each rule.

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