Q1. Solve the following equation by completing the square:

Background

Topic: Quadratic Equations & Completing the Square

This question tests your ability to solve a quadratic equation by the method of completing the square, which is a fundamental algebraic technique used in Precalculus.

Key Terms and Formulas:

• Quadratic equation:

• Completing the square: Transforming into

• Square root property: If , then

Step-by-Step Guidance

1. Start by moving the constant term to the other side:

2. To complete the square, take half of the coefficient of (which is ), square it, and add it to both sides:

3. Add $16y^2 - 8y + 16 = 3 + 16$

4. Rewrite the left side as a perfect square:

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Q2. For the line segment bounded by the given endpoints, and , find the midpoint.

Background

Topic: Coordinate Geometry – Midpoint Formula

This question tests your understanding of how to find the midpoint of a line segment given two endpoints in the coordinate plane.

Key Terms and Formulas:

• Midpoint formula:

• Endpoints: and

Step-by-Step Guidance

1. Identify the coordinates of the endpoints: and

2. Apply the midpoint formula to the -coordinates:

3. Apply the midpoint formula to the -coordinates:

4. Write the midpoint as an ordered pair

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Q3. Solve: and also find domain of $f \circ g(x)$ where ,

Background

Topic: Function Composition and Domain

This question tests your ability to compose two functions and determine the domain of the resulting function.

Key Terms and Formulas:

• Function composition:

• Domain: Set of input values for which the function is defined

• Square root function: is defined for

Step-by-Step Guidance

1. Write as

2. Substitute into :

3. Determine the domain: Since , must be

4. Express the domain in interval notation:

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Q4. Use the standard graph of to transform it into the following function

Background

Topic: Transformations of Absolute Value Functions

This question tests your understanding of how to apply transformations (stretch, shift, translation) to the absolute value function.

Key Terms and Formulas:

• Absolute value function:

• Transformations:

• Vertical stretch: ; Horizontal compression: ; Horizontal shift: ; Vertical shift:

Step-by-Step Guidance

1. Identify the transformations: (vertical stretch), (horizontal compression), (horizontal shift), (vertical shift)

2. Find the vertex: Set

3. The graph is vertically stretched by $2\frac{3}{4}

4. Compare the transformed graph to the options provided

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Q5. Graph as a transformation of the square root function

Background

Topic: Transformations of Square Root Functions

This question tests your ability to graph a transformed square root function and recognize shifts and translations.

Key Terms and Formulas:

• Square root function:

• Transformation:

• Horizontal shift:

Step-by-Step Guidance

1. Set the inside of the square root to zero to find the starting point:

2. The graph of is shifted right by $3f(x)$

3. Compare the transformed graph to the options provided

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Q6. For the following function, sketch its graph:

Background

Topic: Graphing Square Root Functions

This question tests your ability to graph a square root function with a coefficient inside the radical.

Key Terms and Formulas:

• Square root function:

• Domain:

• Transformation: Horizontal stretch/compression

Step-by-Step Guidance

1. Set to find the starting point:

2. For , increases as increases

3. Compare the graph to the options provided

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Q7. Given the two functions and , find and identify the domain.

Background

Topic: Function Operations – Multiplication and Domain

This question tests your ability to multiply two functions and determine the domain of the resulting function.

Key Terms and Formulas:

• Function multiplication:

• Domain: Set of input values for which both functions are defined

Step-by-Step Guidance

1. Write as

2. Substitute the expressions:

3. Expand the product:

4. Simplify the expression and consider the domain (both functions are defined for all real numbers)

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Q8. For the given one-to-one function, find the inverse function, , and verify that your equation is correct.

Background

Topic: Inverse Functions

This question tests your ability to find the inverse of a linear function and verify it by composition.

Key Terms and Formulas:

• Inverse function:

• To find the inverse, solve for

• Verification:

Step-by-Step Guidance

1. Write

2. Solve for in terms of :

3. Replace with to write the inverse:

4. Verify by checking

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Q9. Find and if and

Background

Topic: Inverse Functions and Composition

This question tests your ability to find the inverse of a composition of functions and the composition of their inverses.

Key Terms and Formulas:

• Function composition:

• Inverse of composition:

• Inverse functions: ,

• Composition of inverses:

Step-by-Step Guidance

1. Find :

2. To find , solve for

3. Find and separately

4. Compose

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