Q1. Graph the following piecewise function:

Background

Topic: Piecewise Functions and Graphing

This question tests your understanding of how to interpret and graph piecewise-defined functions, which are functions defined by different expressions depending on the input value of .

Key Terms and Concepts:

• Piecewise Function: A function defined by multiple sub-functions, each applying to a certain interval of the domain.

• Graphing: Plotting each piece on its respective domain interval.

Step-by-Step Guidance

1. Identify the two pieces of the function: for and for .

2. For , plot the line only for values of less than $0$.

3. For , plot the parabola only for values of greater than or equal to $0$.

4. Check the value at for both pieces to determine if there is a jump or if the graph is continuous at .

Try sketching the graph before checking the answer!

Q2. Let , find the following:

Background

Topic: Inverse Functions and Their Properties

This question tests your ability to find the inverse of a function, determine its range, and solve for a specific value.

Key Terms and Formulas:

• Inverse Function: If is a function, is its inverse if .

• Range: The set of all possible output values of a function.

Step-by-Step Guidance

1. To find , set and solve for in terms of .

2. Once you have in terms of , swap and to write the inverse function .

3. To find the range of , consider the domain of the original function and how it maps to the range of the inverse.

4. To solve for such that , set the expression for equal to $1x$.

Try working through the algebra before checking the answer!

Q3. Let

• a. Find

• b. Find the domain and range of

• c. Graph the function

Background

Topic: Logarithmic Functions

This question tests your understanding of evaluating logarithmic functions, determining their domain and range, and graphing them.

Key Terms and Formulas:

• Logarithm: is the exponent to which must be raised to get .

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

• Range: The set of all possible output values.

Step-by-Step Guidance

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

2. For part (b), set to find the domain, and consider the behavior of the logarithm to find the range.

3. For part (c), sketch the graph using key points and the general shape of logarithmic functions.

Try evaluating and sketching before checking the answer!

Q4. Below is the graph of . On the same coordinates, graph the function .

Background

Topic: Graphs of Functions and Their Inverses

This question tests your understanding of how to graph the inverse of a function given its graph.

Key Terms and Concepts:

• Inverse Graph: The graph of is the reflection of across the line .

Step-by-Step Guidance

1. Identify several points on the graph of .

2. For each point on , plot the point for .

3. Draw the line as a reference for reflection.

Try reflecting the points before checking the answer!

Q5. Solve the following exponential equations:

• a.

• b.

• c.

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to solve equations involving exponents and logarithms by applying properties of exponents and logarithms.

Key Terms and Formulas:

• Exponential Equation: An equation where variables appear as exponents.

• Logarithmic Equation: An equation involving logarithms of expressions with variables.

• Key Properties:

  • if and

  • is the natural logarithm (base )

Step-by-Step Guidance

1. For (a), express both sides with the same base and set exponents equal.

2. For (b), divide both sides by $3\ln(x+1)x$.

3. For (c), use the property to combine the logs, then set the arguments equal.

Try solving each equation step-by-step before checking the answer!

Q6. Find the future value if $8,000 is invested for 11 years at interest rate 3% compounded:

• a. Quarterly

• b. Monthly

• c. Continuously

Background

Topic: Compound Interest

This question tests your ability to use compound interest formulas for different compounding periods, including continuous compounding.

Key Terms and Formulas:

• Compound Interest Formula (n times per year):

• Continuous Compounding Formula:

• Where:

  • = future value

  • = principal ()

  • = annual interest rate (as a decimal, )

  • = number of compounding periods per year (quarterly: $4)

  • = number of years ($11$)

Step-by-Step Guidance

1. For (a) and (b), identify , , , and for each case.

2. Plug the values into the compound interest formula for quarterly and monthly compounding.

3. For (c), use the continuous compounding formula and substitute the values for , , and .

4. Set up the expressions for each case, but do not compute the final value yet.

Try plugging in the values and simplifying before checking the answer!