Q1. Find the domain of the following functions. Decide if the graph is even or odd.

Background

Topic: Functions – Domain, Even/Odd Properties

This question tests your understanding of how to determine the set of all possible input values (domain) for a function, and how to classify a function as even, odd, or neither based on its symmetry.

Key Terms and Concepts:

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

• Even Function: for all in the domain.

• Odd Function: for all in the domain.

Step-by-Step Guidance

1. For each function, identify any restrictions on the variable (e.g., denominators cannot be zero, expressions under square roots must be non-negative, etc.).

2. Write the domain in interval notation, considering all restrictions.

3. To determine if the function is even, substitute for and simplify. Compare to .

4. To determine if the function is odd, substitute for and check if .

Try solving on your own before revealing the answer!

Q2. Sketch the graph of for various forms.

Background

Topic: Graphing Functions

This question asks you to sketch the graph of a function for different algebraic forms, which may include transformations such as shifts, stretches, or reflections.

Key Terms and Concepts:

• Parent Function: The basic form of the function (e.g., , ).

• Transformations: Changes to the graph such as vertical/horizontal shifts, stretches, compressions, and reflections.

Step-by-Step Guidance

1. Identify the parent function and note its basic shape.

2. Analyze any coefficients or constants that indicate transformations (e.g., shifts up, shifts right).

3. Apply each transformation step-by-step to the parent graph.

4. Label key points and asymptotes if applicable.

Try sketching before checking the answer!

Q3. Determine if the following function has an inverse. If so, find it.

Background

Topic: Inverse Functions

This question tests your ability to determine if a function is one-to-one (and thus invertible), and to find the inverse function algebraically if it exists.

Key Terms and Formulas:

• One-to-One Function: Each output is produced by exactly one input.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

• Inverse Function: If , then .

Step-by-Step Guidance

1. Check if the function passes the horizontal line test or is strictly increasing/decreasing.

2. If invertible, replace with and solve for in terms of $y$.

3. Swap and to write the inverse function .

4. State the domain and range of the inverse, if required.

Try working through the steps before revealing the answer!

Q4. Simplify the given expressions.

Background

Topic: Algebraic Manipulation

This question tests your ability to simplify algebraic expressions, including rational, exponential, and logarithmic forms.

Key Terms and Formulas:

• Exponent Rules: ,

• Logarithm Rules: ,

Step-by-Step Guidance

1. Apply exponent and logarithm rules as appropriate to combine or simplify terms.

2. Factor or expand expressions where possible to further simplify.

3. Check for restrictions on the variable (e.g., for ).

Try simplifying before checking the answer!

Q5. For each of the following limits, decide if the limit exists or does not exist. If it does exist, find it. If it does not exist, explain or show why. (Do not use L'Hôpital's Rule.)

Background

Topic: Limits and Continuity

This question tests your understanding of how to evaluate limits using algebraic manipulation, factoring, rationalization, and recognizing indeterminate forms.

Key Terms and Formulas:

• Limit: is the value approaches as approaches .

• Indeterminate Forms: , , etc.

• Factoring and Rationalization: Useful for simplifying expressions to resolve indeterminate forms.

Step-by-Step Guidance

1. Substitute the value into the function to check if you get a determinate or indeterminate form.

2. If indeterminate, factor numerator and denominator or rationalize as needed.

3. Simplify the expression and try substituting again.

4. If the limit does not exist, explain why (e.g., division by zero, oscillation, etc.).

Try evaluating the limits before revealing the answer!

Q6. Find all values of (and if it exists) such that is continuous at .

Background

Topic: Continuity of Piecewise Functions

This question tests your ability to determine the values of parameters that make a piecewise function continuous at a given point.

Key Terms and Formulas:

• Continuity at :

Step-by-Step Guidance

1. Write the left-hand and right-hand limits of as approaches .

2. Set the left-hand limit equal to the right-hand limit and to .

3. Solve for (and if needed) to ensure continuity at .

Try setting up the equations before revealing the answer!

Q7. Decide whether the function has any vertical or horizontal asymptotes. Explain your answer.

Background

Topic: Asymptotes of Rational Functions

This question tests your ability to find vertical and horizontal asymptotes by analyzing the behavior of rational functions as approaches certain values or infinity.

Key Terms and Formulas:

• Vertical Asymptote: Occurs where the denominator is zero and the numerator is not zero.

• Horizontal Asymptote: Determined by the degrees of the numerator and denominator as .

Step-by-Step Guidance

1. Set the denominator equal to zero to find possible vertical asymptotes.

2. Compare the degrees of the numerator and denominator to determine horizontal asymptotes.

3. Check for any simplification that might remove an asymptote (hole).

Try identifying the asymptotes before revealing the answer!

Q8. Use the limit definition of derivative to find , if it exists.

Background

Topic: Definition of the Derivative

This question tests your ability to use the formal limit definition to compute the derivative of a function.

Key Formula:

Step-by-Step Guidance

1. Write out and for the given function.

2. Substitute into the limit definition formula.

3. Simplify the numerator algebraically.

4. Factor and cancel if possible, then take the limit as .

Try applying the definition before revealing the answer!

Q9. Use the limit definition of derivative to find at a specific value.

Background

Topic: Derivative at a Point

This question tests your ability to use the limit definition to find the derivative at a specific value .

Key Formula:

Step-by-Step Guidance

1. Compute and for the given function and value .

2. Substitute into the limit definition formula.

3. Simplify the numerator and denominator as much as possible.

4. Take the limit as to find the derivative at .

Try working through the algebra before revealing the answer!

Q10. Find the second derivative of the function.

Background

Topic: Higher-Order Derivatives

This question tests your ability to compute the second derivative, which is the derivative of the first derivative.

Key Formula:

Step-by-Step Guidance

1. Find the first derivative using standard differentiation rules.

2. Differentiate to obtain .

3. Simplify the result as much as possible.

Try differentiating before revealing the answer!

Q11. Find the second derivative at the indicated point.

Background

Topic: Evaluating Higher-Order Derivatives at a Point

This question tests your ability to compute the second derivative and evaluate it at a specific value of .

Key Formula:

• means find and then substitute .

Step-by-Step Guidance

1. Find and then as in the previous question.

2. Substitute the given value of into .

3. Simplify the expression to get the value at the point.

Try evaluating before revealing the answer!

Q12. Find the equation of the tangent line to the graph of at a given point.

Background

Topic: Tangent Lines and Derivatives

This question tests your ability to find the equation of the tangent line to a function at a specific point using derivatives.

Key Formula:

• The tangent line at :

Step-by-Step Guidance

1. Find , the derivative of the function.

2. Evaluate at the given -value to find the slope of the tangent line.

3. Use the point-slope form with the given point to write the equation of the tangent line.

Try setting up the tangent line before revealing the answer!

Q13. Find the first derivatives of each of the following functions. Simplify your final answer.

Background

Topic: Differentiation Rules

This question tests your ability to apply the power rule, product rule, quotient rule, and chain rule to find derivatives of various functions.

Key Formulas:

• Power Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Step-by-Step Guidance

1. Identify which differentiation rule(s) apply to each function.

2. Apply the rule(s) step-by-step, simplifying as you go.

3. Rewrite the derivative without negative exponents or complex fractions, as requested.

Try differentiating before revealing the answer!