Q1. Determine (f ∘ g)(-1), where f(x) = 25 - x2 and g(x) = x2 - 4x.

Background

Topic: Function Composition

This question tests your understanding of how to compose two functions and evaluate the result at a specific input.

Key Terms and Formulas:

• Function composition:

• Evaluate , then use that value as the input for .

Step-by-Step Guidance

1. First, calculate by substituting into .

2. Take the result from and substitute it into to find .

3. Set up the expression for and simplify as much as possible.

Try solving on your own before revealing the answer!

Q2. Using the Rational-Root Theorem, list all the possible rational zeros of the given polynomial function, points will be deducted for listing incorrect answers.

Background

Topic: Rational Root Theorem

This question tests your ability to apply the Rational Root Theorem to a polynomial to list all possible rational zeros.

Key Terms and Formulas:

• Rational Root Theorem: Possible rational roots are , where is a factor of the constant term and is a factor of the leading coefficient.

Step-by-Step Guidance

1. Identify the constant term and the leading coefficient in the polynomial.

2. List all integer factors of the constant term () and the leading coefficient ().

3. Form all possible fractions using the factors found.

Try solving on your own before revealing the answer!

Q3. Analytically solve for the exact real solution of the following equation:

Background

Topic: Logarithmic Equations

This question tests your ability to solve equations involving logarithms and exponentials.

Key Terms and Formulas:

• Logarithmic properties:

Step-by-Step Guidance

1. Rewrite the equation in exponential form:

2. Solve for by isolating $x$ in the equation.

3. Check the solution to ensure it is a real number and satisfies the original equation.

Try solving on your own before revealing the answer!

Q4. Graph the following function. Draw and label its asymptote and 2 points on the graph:

Background

Topic: Logarithmic Functions and Graphing

This question tests your ability to graph logarithmic functions, identify asymptotes, and plot points.

Key Terms and Formulas:

• Vertical asymptote: Occurs where the argument of the logarithm is zero.

• Domain:

• Range: All real numbers

Step-by-Step Guidance

1. Identify the vertical asymptote by setting .

2. Choose two values of greater than 1, substitute into , and calculate the corresponding values.

3. Sketch the graph, labeling the asymptote and the two points.

Try solving on your own before revealing the answer!

Q5. Graph the parabola. Find its vertex, focus, endpoints of its latus rectum, and directrix:

Background

Topic: Parabolas and Conic Sections

This question tests your ability to analyze and graph parabolas, and find key features such as vertex, focus, latus rectum, and directrix.

Key Terms and Formulas:

• Standard form for a parabola:

• Vertex:

• Focus:

• Directrix:

• Latus rectum endpoints:

Step-by-Step Guidance

1. Rewrite the equation in standard form and identify , , and .

2. Find the vertex, focus, and directrix using the formulas above.

3. Calculate the endpoints of the latus rectum.

Try solving on your own before revealing the answer!