Precalculus Exam 2: Step-by-Step Study Guidance

Q1. Let $f(x) = -x^2 - x$. Find $f(-a)$.

Background

Topic: Function Evaluation and Substitution

This question tests your ability to substitute a value (in this case, $-a$) into a function and simplify the result.

Key Terms and Formulas

• Function evaluation: To find $f(k)$, replace every $x$ in $f(x)$ with $k$.

• Squaring a negative: $(-a)^2 = a^2$.

Step-by-Step Guidance

1. Start by writing the function: $f(x) = -x^2 - x$.

2. Substitute $x = -a$ into the function: $f(-a) = -(-a)^2 - (-a)$.

3. Simplify $(-a)^2$ and $-(-a)$ separately.

4. Combine the simplified terms to get the final expression for $f(-a)$.

Try solving on your own before revealing the answer!

Final Answer: D. $-a^2 - a$

Substituting $-a$ into the function gives $-(-a)^2 - (-a) = -a^2 + a$. But since the function is $-x^2 - x$, the correct simplification is $-a^2 - a$.

Q2. Find the $x$-intercepts for $g(x) = x^3 - x$.

Background

Topic: Finding $x$-intercepts (Roots/Zeros) of a Function

This question tests your ability to find the values of $x$ where the function equals zero.

Key Terms and Formulas

• $x$-intercept: A point where $y = 0$, so solve $g(x) = 0$.

• Factoring: $x^3 - x = x(x^2 - 1) = x(x-1)(x+1)$.

Step-by-Step Guidance

1. Set $g(x) = 0$: $x^3 - x = 0$.

2. Factor the equation completely.

3. Set each factor equal to zero and solve for $x$.

4. Write the $x$-intercepts as points $(x, 0)$.

Try solving on your own before revealing the answer!

Final Answer: B. III (0, 0), (1, 0), and (−1, 0)

Factoring gives $x(x-1)(x+1) = 0$, so $x = 0, 1, -1$ are the $x$-intercepts.

Q3. Which of the following are functions?

Background

Topic: Definition of a Function (Vertical Line Test, Uniqueness of Output)

This question tests your understanding of what makes a relation a function: each input $x$ must correspond to exactly one output $y$.

Key Terms and Formulas

• Function: For every $x$, there is only one $y$.

• Vertical Line Test: A graph is a function if no vertical line crosses it more than once.

Step-by-Step Guidance

1. Analyze each equation to see if, for every $x$, there is only one $y$.

2. For equations involving $y^2$, consider if a single $x$ can yield more than one $y$.

3. For equations solved for $y$, check if $y$ is uniquely determined by $x$.

4. List which options meet the function criteria.

Try solving on your own before revealing the answer!

Final Answer: B. I and III only

Only I and III define $y$ uniquely for each $x$; II and IV do not because they involve $y^2$ or $x^2$ terms.

Q4. Suppose we perform the following transformations to the graph of $f$ in the indicated order: 1. reflection over the $y$-axis, 2. vertical stretch by a factor of 3, 3. shift down 2 units. What is the equation of the transformed graph?

Background

Topic: Transformations of Functions

This question tests your understanding of how to apply multiple transformations to a function and write the resulting equation.

Key Terms and Formulas

• Reflection over $y$-axis: Replace $x$ with $-x$.

• Vertical stretch by $a$: Multiply the function by $a$.

• Shift down $k$ units: Subtract $k$ from the function.

Step-by-Step Guidance

1. Start with $f(x)$.

2. Apply the reflection over the $y$-axis: $f(-x)$.

3. Apply the vertical stretch: $3f(-x)$.

4. Apply the downward shift: $3f(-x) - 2$.

Try solving on your own before revealing the answer!

Final Answer: A. $g(x) = 3f(-x) - 2$

Each transformation is applied in the correct order, resulting in $3f(-x) - 2$.

Q5. Suppose we perform the following transformations to the graph of $f(x) = |x|$ in the indicated order: 1. shift left 2 units, 2. reflection over the $x$-axis, 3. vertical stretch by a factor of 3, 4. shift up 1 unit. What is the equation of the transformed graph?

Background

Topic: Transformations of Absolute Value Functions

This question tests your ability to apply multiple transformations to the absolute value function and write the resulting equation.

Key Terms and Formulas

• Shift left $h$ units: Replace $x$ with $x + h$.

• Reflection over $x$-axis: Multiply the function by $-1$.

• Vertical stretch by $a$: Multiply the function by $a$.

• Shift up $k$ units: Add $k$ to the function.

Step-by-Step Guidance

1. Start with $f(x) = |x|$.

2. Shift left 2 units: $|x + 2|$.

3. Reflect over the $x$-axis: $-|x + 2|$.

4. Vertical stretch by 3: $-3|x + 2|$.

5. Shift up 1 unit: $-3|x + 2| + 1$.

Try solving on your own before revealing the answer!

Final Answer: D. $g(x) = -3|x + 2| + 1$

Each transformation is applied in the correct order, resulting in $-3|x + 2| + 1$.

Q6. Which of the following points on a graph are NOT moved by a horizontal reflection? Select all that apply.

Background

Topic: Symmetry and Transformations

This question tests your understanding of how points are affected by a reflection over the $y$-axis (horizontal reflection).

Key Terms and Formulas

• Horizontal reflection: A point $(x, y)$ becomes $(-x, y)$ after reflection over the $y$-axis.

• Invariant points: Points where $x = 0$ are unchanged by this reflection.

Step-by-Step Guidance

1. For each point, apply the transformation $(x, y) \to (-x, y)$.

2. Check which points remain unchanged after this transformation.

3. Identify all such points from the list.

Try solving on your own before revealing the answer!

Final Answer: B and D. (0, 1) and (0, 0)

Points with $x = 0$ are not moved by a horizontal reflection.

Q7. What is the domain of $f(x) = \frac{x}{\sqrt{1 - x}}$?

Background

Topic: Domain of Functions Involving Square Roots and Fractions

This question tests your ability to find the domain of a function that includes a square root in the denominator.

Key Terms and Formulas

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

• Square root in denominator: The expression under the square root must be positive (not zero or negative).

Step-by-Step Guidance

1. Set the denominator $\sqrt{1 - x}$ to be greater than zero: $1 - x > 0$.

2. Solve the inequality for $x$.

3. Express the solution in interval notation.

Try solving on your own before revealing the answer!

Final Answer: C. $(-\infty, 1)$

The denominator is defined and positive only when $x < 1$.

Q8. Let $f(x) = x^2$ and $g(x) = \sqrt{x - 1}$. What is the domain of $f \circ g$?

Background

Topic: Domain of Composite Functions

This question tests your ability to find the domain of a composite function, considering the domains of both $f$ and $g$.

Key Terms and Formulas

• Composite function: $(f \circ g)(x) = f(g(x))$.

• Domain of $g(x)$: $x - 1 \geq 0$.

• Domain of $f(x)$: All real numbers.

Step-by-Step Guidance

1. Find the domain of $g(x) = \sqrt{x - 1}$ by setting $x - 1 \geq 0$.

2. Solve for $x$ to find the domain of $g$.

3. Since $f(x) = x^2$ is defined for all real numbers, the domain of $f \circ g$ is the same as the domain of $g$.

Try solving on your own before revealing the answer!

Final Answer: D. $[1, \infty)$

The composite function is defined for $x \geq 1$.

Q9. Let $f(x) = \frac{x}{x-1}$ and $g(x) = \frac{2}{x}$. Find $(f \circ g)(x)$ and simplify.

Background

Topic: Composition of Rational Functions

This question tests your ability to compose two rational functions and simplify the result.

Key Terms and Formulas

• Composition: $(f \circ g)(x) = f(g(x))$.

• Substitution: Replace $x$ in $f(x)$ with $g(x)$.

Step-by-Step Guidance

1. Write $f(g(x)) = \frac{g(x)}{g(x) - 1}$.

2. Substitute $g(x) = \frac{2}{x}$ into $f$.

3. Simplify the numerator and denominator.

4. Combine terms to get a single rational expression.

Try solving on your own before revealing the answer!

Final Answer: B. $\frac{2/x}{2/x - 1} = \frac{2}{2 - x}$

After substitution and simplification, the result is $\frac{2}{2 - x}$.

Q10. Find the inverse of $f(x) = \frac{2}{x + 3}$.

Background

Topic: Inverse Functions

This question tests your ability to find the inverse of a rational function by solving for $x$ in terms of $y$ and then switching variables.

Key Terms and Formulas

• Inverse function: If $y = f(x)$, solve for $x$ in terms of $y$, then replace $y$ with $x$ to get $f^{-1}(x)$.

Step-by-Step Guidance

1. Set $y = \frac{2}{x + 3}$.

2. Multiply both sides by $(x + 3)$ to clear the denominator.

3. Solve for $x$ in terms of $y$.

4. Switch $x$ and $y$ to write the inverse function $f^{-1}(x)$.

Try solving on your own before revealing the answer!

Final Answer: C. $f^{-1}(x) = \frac{2}{x} - 3$

Solving for $x$ and switching variables gives the inverse function.