Q1. Let be given. Find the graph of the inverse function of .

Background

Topic: Inverse Functions and Exponential Functions

This question tests your understanding of how to find the inverse of an exponential function and how to graph it.

Key Terms and Formulas:

• Inverse function: If is a function, its inverse satisfies .

• Exponential function: is the base- exponential function.

• To find the inverse, solve for in terms of .

Step-by-Step Guidance

1. Write the function as .

2. Isolate by subtracting 2 from both sides: .

3. Take the natural logarithm of both sides to solve for : .

4. Express the inverse function: .

5. Think about the domain of the inverse function: For , .

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Q2. Find the domain of the function .

Background

Topic: Domain of Rational Functions

This question tests your ability to determine the domain of a rational function by identifying values that make the denominator zero.

Key Terms and Formulas:

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

• Rational function: A function of the form .

• Denominator cannot be zero: .

Step-by-Step Guidance

1. Set the denominator equal to zero: .

2. Expand to .

3. Rewrite the equation: .

4. Solve for to find values that make the denominator zero.

5. The domain excludes these values.

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Q3. Find the domain of the function .

Background

Topic: Domain of Logarithmic Functions

This question tests your ability to find the domain of a logarithmic function, which requires the argument to be positive.

Key Terms and Formulas:

• Logarithmic function: is defined only for .

• Argument: .

Step-by-Step Guidance

1. Set the argument greater than zero: .

2. Analyze when the fraction is positive: and , or and .

3. Since , gives .

4. Consider if there are any other intervals where the argument is positive.

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Q4. A rectangular storage container with an open top has a volume of . The length of its base is twice its width. Material for the base costs $10 per square meter. Express the cost of materials as a function of the width of the base.

Background

Topic: Optimization and Cost Functions

This question tests your ability to express a cost function in terms of a variable, using geometric relationships and algebraic manipulation.

Key Terms and Formulas:

• Volume of a box: .

• Area of base: .

• Area of sides: .

• Cost: .

Step-by-Step Guidance

1. Let the width be , so the length is .

2. Let the height be . The volume is .

3. Solve for in terms of : .

4. Calculate the area of the base: .

5. Calculate the area of the sides: .

6. Express the total cost as a function of using the formulas above.

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Q5. The population of a certain country (in millions) at time (in years) is modeled as , where is the year 2000. When will the population reach 12 million?

Background

Topic: Exponential Growth and Solving Equations

This question tests your ability to solve exponential equations for a given value.

Key Terms and Formulas:

• Exponential growth: .

• To solve for , set and solve for .

Step-by-Step Guidance

1. Set : .

2. Divide both sides by 4: .

3. Take the natural logarithm of both sides: .

4. Solve for by dividing both sides by 0.05.

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Q6. Determine the period, maximum, and minimum value of the sinusoidal function .

Background

Topic: Sinusoidal Functions

This question tests your understanding of the properties of sine functions, including amplitude, period, and vertical shift.

Key Terms and Formulas:

• General form: .

• Amplitude: .

• Period: .

• Vertical shift: .

Step-by-Step Guidance

1. Identify amplitude: .

2. Identify period: .

3. Identify vertical shift: $3$.

4. Maximum value: .

5. Minimum value: .

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Q7. Build a sinusoidal formula for , the number of daylight hours as a function of , where is the number of days after January 1. The number of sunlight hours oscillates between 9.5 and 14.5 on the yearly cycle (1 year = 364 days). The longest day is the summer solstice, which is 170 days after January 1.

Background

Topic: Sinusoidal Modeling

This question tests your ability to model periodic phenomena using sinusoidal functions.

Key Terms and Formulas:

• General form: .

• Amplitude: .

• Vertical shift: .

• Period: days.

• Phase shift: is the day of maximum.

Step-by-Step Guidance

1. Calculate amplitude: .

2. Calculate vertical shift: .

3. Period is $364\frac{2\pi}{364}$.

4. Phase shift is $170t = 170$).

5. Write the formula: .

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Q8. What is the domain of the radical function ?

Background

Topic: Domain of Radical Functions

This question tests your ability to find the domain of a function involving a square root, which requires the argument to be non-negative.

Key Terms and Formulas:

• Radical function: is defined only when .

• Set .

Step-by-Step Guidance

1. Set the argument of the square root greater than or equal to zero: .

2. Solve for : .

3. Divide both sides by (remember to reverse the inequality): .

4. Write the domain in interval notation.

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Q9. Unit Circle: Reference for Trigonometric Functions

Background

Topic: Trigonometric Functions and the Unit Circle

The unit circle is a fundamental tool for understanding the values of sine, cosine, and tangent at various angles, and is essential for solving trigonometric equations.

Key Terms and Formulas:

• Unit circle: A circle of radius 1 centered at the origin.

• Coordinates correspond to for angle .

• Common angles: , etc.

Step-by-Step Guidance

1. Use the unit circle to find sine, cosine, and tangent values for common angles.

2. Apply these values when solving trigonometric equations or evaluating functions.

3. Remember the symmetry and periodicity of trigonometric functions.

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