Q1. Given , , and , find the following:

• a.

• b.

• c.

• d.

• e.

• f.

Background

Topic: Function Composition

This question tests your understanding of how to compose functions and evaluate them at specific values.

Key Terms and Formulas:

• Function composition:

• Evaluate functions by substituting values or expressions into the function definitions.

Step-by-Step Guidance

1. Start by evaluating the innermost function for each part. For example, in part (a), compute first.

2. Once you have the result from the inner function, substitute it into the outer function. For part (a), use the value from as the input for .

3. For parts (c) and (d), substitute the entire function expression (not just a number) into the other function. For example, means replace in with .

4. For part (e), compose three functions: first, then applied to $h(x)$, then applied to .

5. For part (f), use and follow the same process as in part (e), but with a specific value.

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Q2. Find the natural domains and ranges of the following expressions:

• a.

• b.

Background

Topic: Domain and Range

This question tests your ability to determine the set of input values (domain) and output values (range) for given functions, especially those involving square roots and rational expressions.

Key Terms and Formulas:

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

• Range: The set of all possible output values.

• For , .

• For , .

Step-by-Step Guidance

1. For part (a), identify restrictions from the square root: .

2. Also, the denominator cannot be zero, so set and solve for .

3. For part (b), again, requires . There are no other restrictions since the expression is defined for all $x \geq 0$.

4. To find the range, consider the possible values the expressions can take as varies over the domain.

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Q3. If and , write a formula for in terms of .

Background

Topic: Function Composition

This question tests your ability to compose two functions and write the resulting formula.

Key Terms and Formulas:

• Function composition:

Step-by-Step Guidance

1. Start by finding , which is .

2. Substitute into : .

3. Replace with in the formula.

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Q4. Find the natural domain and range of your answer to question 3.

Background

Topic: Domain and Range of Composite Functions

This question tests your ability to determine the domain and range of a function involving a square root.

Key Terms and Formulas:

• For , .

• Domain: Set and solve for .

• Range: Consider the possible values of as varies over the domain.

Step-by-Step Guidance

1. Set to find the domain.

2. Solve for to get the interval of allowed values.

3. For the range, consider the minimum and maximum values of as varies within the domain.

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Q5. Determine whether the following graphs represent functions.

Background

Topic: Function Identification from Graphs

This question tests your ability to use the vertical line test to determine if a graph represents a function.

Key Terms and Formulas:

• Vertical Line Test: If any vertical line crosses the graph more than once, it is not a function.

Step-by-Step Guidance

1. Examine each graph and imagine drawing vertical lines at various points along the -axis.

2. If a vertical line crosses the graph at more than one point, the graph does not represent a function.

3. If every vertical line crosses the graph at most once, it is a function.

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Q6. Express the area of a circle in terms of its circumference .

Background

Topic: Geometric Formulas and Variable Substitution

This question tests your ability to relate the area and circumference of a circle using algebraic manipulation.

Key Terms and Formulas:

• Area:

• Circumference:

Step-by-Step Guidance

1. Start with the formula for circumference: .

2. Solve for in terms of .

3. Substitute this expression for into the area formula.

4. Simplify the resulting expression to write area in terms of circumference .

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Q7. Express a cube’s central diagonal length in terms of its surface area .

Background

Topic: Geometric Relationships

This question tests your ability to relate the diagonal of a cube to its surface area using algebraic manipulation.

Key Terms and Formulas:

• Surface area of a cube:

• Central diagonal (space diagonal):

Step-by-Step Guidance

1. Start with the surface area formula: .

2. Solve for in terms of .

3. Substitute this value of into the diagonal formula.

4. Simplify to express the diagonal in terms of surface area .

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Q8. Graph the following functions:

• a.

• b.

Background

Topic: Graphing Quadratic and Radical Functions

This question tests your ability to graph quadratic and radical functions, including understanding their domain and range.

Key Terms and Formulas:

• Quadratic function:

• Radical function:

Step-by-Step Guidance

1. For part (a), identify the vertex, axis of symmetry, and intercepts of the quadratic function.

2. For part (b), determine the domain by setting and solve for .

3. Plot key points and sketch the general shape of each graph.

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Q9. Graph the piecewise function

Background

Topic: Piecewise Functions

This question tests your ability to graph functions defined by different expressions over different intervals.

Key Terms and Formulas:

• Piecewise function: Defined by different formulas for different intervals of .

Step-by-Step Guidance

1. Identify the intervals for each piece: and .

2. For each interval, plot the corresponding linear function.

3. Check endpoints to determine if they are included or excluded (closed or open circles).

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Q10. Determine the piecewise function depicted in the graph.

Background

Topic: Interpreting Piecewise Functions from Graphs

This question tests your ability to write a piecewise function based on a given graph.

Key Terms and Formulas:

• Piecewise function: Different formulas for different intervals.

Step-by-Step Guidance

1. Analyze the graph to identify intervals and the corresponding expressions.

2. Write the function for each interval, noting where the function changes.

3. Check endpoints for inclusion/exclusion.

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Q11. Determine whether the following functions are even, odd, or neither:

• a.

• b.

• c.

Background

Topic: Function Symmetry

This question tests your ability to determine whether a function is even, odd, or neither by analyzing its algebraic form.

Key Terms and Formulas:

• Even function: for all in the domain.

• Odd function: for all in the domain.

Step-by-Step Guidance

1. For each function, substitute for and simplify.

2. Compare the result to the original function to see if it matches (even), is the negative (odd), or neither.

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Q12. The graphs of , , and are horizontal and vertical shifts of . Determine , , and .

Background

Topic: Shifting Quadratic Functions

This question tests your ability to identify horizontal and vertical shifts in quadratic functions based on their graphs.

Key Terms and Formulas:

• General form:

• Horizontal shift:

• Vertical shift:

Step-by-Step Guidance

1. Identify the vertex of each shifted parabola from the graph.

2. Write the equation using the vertex form, substituting the values for and .

3. Check the direction of the shift (left/right/up/down).

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Q13. The graphs of and are horizontal and vertical scale transformations of . Determine and .

Background

Topic: Transformations of Radical Functions

This question tests your ability to identify scale transformations in radical functions based on their graphs.

Key Terms and Formulas:

• General form:

• Vertical scale:

• Horizontal scale:

Step-by-Step Guidance

1. Analyze the graph to determine how the function has been stretched or compressed vertically or horizontally.

2. Write the equation using the appropriate scale factors.

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Q14. Graph the following functions:

• a.

• b.

Background

Topic: Graphing Rational and Radical Functions

This question tests your ability to graph rational and radical functions, including understanding their domain and range.

Key Terms and Formulas:

• Rational function:

• Radical function:

Step-by-Step Guidance

1. For part (a), identify the domain (where denominator is not zero) and asymptotes.

2. For part (b), set to find the domain.

3. Plot key points and sketch the general shape of each graph.

Try solving on your own before revealing the answer!