College Algebra Practice Final Exam Step-by-Step Guidance

Notes Practice Video lessons

Study Guide - Smart Notes

Tailored notes based on your materials, expanded with key definitions, examples, and context.

Q1. Evaluate $f(-1)$ for $f(x) = 7x^2 - 3x + 2$.

Background

Topic: Function Evaluation

This question tests your ability to substitute a value into a function and simplify the result.

Key Terms and Formulas:

Function: $f(x)$ is a rule that assigns each input $x$ to exactly one output.
Substitution: Replace $x$ with the given value.

Step-by-Step Guidance

Write the function: $f(x) = 7x^2 - 3x + 2$.
Substitute $x = -1$ into the function: $f(-1) = 7(-1)^2 - 3(-1) + 2$.
Calculate $(-1)^2$ and multiply by 7.
Multiply $-3$ by $-1$.
Add all terms together to get the result.

Try solving on your own before revealing the answer!

Final Answer: $f(-1) = 12$

Substituting $x = -1$ gives $f(-1) = 7(1) + 3 + 2 = 12$.

Q2. Determine the domain and range of the relation. Is it a function? Is it one-to-one?

Background

Topic: Relations and Functions

This question tests your understanding of domain, range, and the definition of a function and one-to-one function.

Key Terms:

Domain: Set of all possible input values.
Range: Set of all possible output values.
Function: Each input has exactly one output.
One-to-one: Each output is paired with only one input.

Step-by-Step Guidance

Identify the set of inputs (domain) and outputs (range) from the relation.
Check if each input is paired with only one output to determine if it is a function.
Check if each output is paired with only one input to determine if it is one-to-one.
Write the domain and range as lists or sets.

Try solving on your own before revealing the answer!

Final Answer:

Domain: Countries (Belgium, Kenya, Norway, Thailand) Range: Continents (Asia, Africa, Europe) Function? Yes One-to-one? No (since more than one country can be in the same continent)

Q3. Determine the domain and range of the relation. Is it a function? Is it one-to-one?

Background

Topic: Relations and Functions

This question is similar to Q2 and tests your understanding of domain, range, and function properties.

Key Terms:

Domain: Set of all possible input values.
Range: Set of all possible output values.
Function: Each input has exactly one output.
One-to-one: Each output is paired with only one input.

Step-by-Step Guidance

List the domain and range based on the given relation.
Check if each input is paired with only one output.
Check if each output is paired with only one input.
Write your answers for domain, range, function, and one-to-one.

Try solving on your own before revealing the answer!

Final Answer:

Domain: Belgium, Kenya, Norway, Thailand Range: Asia, Africa, Europe Function? Yes One-to-one? No

Q4. Sketch the graph of the equation $x = 4$.

Background

Topic: Graphing Linear Equations

This question tests your ability to graph a vertical line.

Key Terms:

Vertical line: All points have the same $x$-value.

Step-by-Step Guidance

Recognize that $x = 4$ is a vertical line.
On a coordinate plane, plot the line where every point has $x = 4$.
The line will pass through points like $(4, y)$ for any $y$.

Try sketching the graph before revealing the answer!

Final Answer:

The graph is a vertical line passing through $x = 4$.

Q5. Sketch the graph of the equation $y = -\frac{5}{6}x + 4$.

Background

Topic: Graphing Linear Equations

This question tests your ability to graph a line in slope-intercept form.

Key Terms and Formulas:

Slope-intercept form: $y = mx + b$
Slope ($m$): $-\frac{5}{6}$
Y-intercept ($b$): $4$

Step-by-Step Guidance

Identify the slope and y-intercept from the equation.
Plot the y-intercept at $(0, 4)$ on the graph.
Use the slope to find another point: from $(0, 4)$, move down 5 units and right 6 units.
Draw a straight line through these points.

Try sketching the graph before revealing the answer!

Final Answer:

The graph is a straight line with slope $-\frac{5}{6}$ and y-intercept $4$.

Q6. Find the slope-intercept form of the line passing through $(6, -3)$ and $(3, 2)$.

Background

Topic: Linear Equations

This question tests your ability to find the equation of a line given two points.

Key Terms and Formulas:

Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Slope-intercept form: $y = mx + b$

Step-by-Step Guidance

Label the points: $(x_1, y_1) = (6, -3)$ and $(x_2, y_2) = (3, 2)$.
Calculate the slope: $m = \frac{2 - (-3)}{3 - 6}$.
Simplify the slope calculation.
Use the slope and one point to solve for $b$ in $y = mx + b$.
Write the final equation in slope-intercept form.

Try solving on your own before revealing the answer!

Final Answer:

Slope $m = -1.67$, y-intercept $b = 7$; equation: $y = -1.67x + 7$.

Q7. Solve the inequality $3(x-3) + 1 > 5(x-2)$ and write your answer in interval notation.

Background

Topic: Linear Inequalities

This question tests your ability to solve linear inequalities and express the solution in interval notation.

Key Terms and Formulas:

Linear inequality: An inequality involving a linear expression.
Interval notation: A way to express the set of solutions.

Step-by-Step Guidance

Expand both sides: $3(x-3) + 1$ and $5(x-2)$.
Simplify each side.
Move all terms involving $x$ to one side and constants to the other.
Solve for $x$.
Express the solution in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

$x < 2$; interval notation: $(-\infty, 2)$

Q8. State the open intervals over which the function is increasing, decreasing, and constant. On what open intervals is the graph continuous?

Background

Topic: Function Behavior

This question tests your ability to analyze a function's graph for intervals of increase, decrease, constancy, and continuity.

Key Terms:

Increasing: Function values rise as $x$ increases.
Decreasing: Function values fall as $x$ increases.
Constant: Function values stay the same.
Continuous: No breaks or jumps in the graph.

Step-by-Step Guidance

Examine the graph to identify where the function rises, falls, or stays flat.
Write the intervals for each behavior.
Identify where the graph is continuous (no gaps).

Try analyzing the graph before revealing the answer!

Final Answer:

Answers depend on the specific graph provided. Typically, increasing, decreasing, and constant intervals are written as open intervals, and continuity is described as the union of intervals where the graph has no breaks.

Q9. Use transformations to sketch the graph of $f(x) = |x-3| - 4$. State the domain and range.

Background

Topic: Absolute Value Functions and Transformations

This question tests your understanding of graph transformations for absolute value functions.

Key Terms and Formulas:

Absolute value function: $f(x) = |x|$
Transformation: Shifts and translations of the graph.

Step-by-Step Guidance

Recognize the parent function $|x|$.
Identify the horizontal shift: $x-3$ shifts the graph right by 3 units.
Identify the vertical shift: $-4$ shifts the graph down by 4 units.
Sketch the new vertex at $(3, -4)$.
Determine the domain and range based on the graph.

Try sketching and stating domain/range before revealing the answer!

Final Answer:

Domain: $(-\infty, \infty)$ Range: $[-4, \infty)$ The graph is a 'V' shape with vertex at $(3, -4)$.

Q10. Use transformations to sketch the graph of $f(x) = -2(x+1)^3 - 3$. State the domain and range.

Background

Topic: Cubic Functions and Transformations

This question tests your understanding of graph transformations for cubic functions.

Key Terms and Formulas:

Cubic function: $f(x) = x^3$
Transformation: Shifts, stretches, and reflections.

Step-by-Step Guidance

Recognize the parent function $x^3$.
Identify the horizontal shift: $x+1$ shifts the graph left by 1 unit.
Identify the vertical shift: $-3$ shifts the graph down by 3 units.
Identify the vertical stretch and reflection: $-2$ multiplies the graph by $-2$ (stretches and reflects).
Sketch the new inflection point at $(-1, -3)$.
Determine the domain and range based on the graph.

Try sketching and stating domain/range before revealing the answer!

Final Answer:

Domain: $(-\infty, \infty)$ Range: $(-\infty, \infty)$ The graph is a cubic curve, reflected and stretched, with inflection at $(-1, -3)$.

Q11. Solve the inequality $|2x+1| \leq 9$ and write your answer in interval notation.

Background

Topic: Absolute Value Inequalities

This question tests your ability to solve inequalities involving absolute values.

Key Terms and Formulas:

Absolute value inequality: $|A| \leq B$ means $-B \leq A \leq B$.
Interval notation: Expressing the solution as a range.

Step-by-Step Guidance

Rewrite the inequality: $-9 \leq 2x+1 \leq 9$.
Subtract 1 from all parts: $-10 \leq 2x \leq 8$.
Divide all parts by 2: $-5 \leq x \leq 4$.
Express the solution in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

Interval notation: $[-5, 4]$

Q12. Use completing the square to rewrite $f(x) = 2x^2 - 12x + 12$ for graphing. Then answer: domain, range, vertex, axis of symmetry, max/min value, and intervals of increase/decrease.

Background

Topic: Quadratic Functions and Completing the Square

This question tests your ability to rewrite a quadratic in vertex form and analyze its graph.

Key Terms and Formulas:

Completing the square: Rewriting $ax^2 + bx + c$ as $a(x-h)^2 + k$.
Vertex form: $f(x) = a(x-h)^2 + k$
Vertex: $(h, k)$
Axis of symmetry: $x = h$
Domain: $(-\infty, \infty)$
Range: Depends on $a$ and $k$

Step-by-Step Guidance

Factor out $2$ from the first two terms: $f(x) = 2(x^2 - 6x) + 12$.
Complete the square inside the parentheses: $x^2 - 6x$.
Add and subtract the necessary value to complete the square.
Rewrite the function in vertex form.
Identify the vertex, axis of symmetry, domain, range, and intervals of increase/decrease.

Try completing the square and analyzing before revealing the answer!

Final Answer:

Vertex form: $f(x) = 2(x-3)^2 - 6$ Domain: $(-\infty, \infty)$ Range: $[-6, \infty)$ Vertex: $(3, -6)$ Axis of symmetry: $x = 3$ Minimum value: $-6$ Increasing: $(3, \infty)$ Decreasing: $(-\infty, 3)$

Q13. Solve the quadratic equation $x^2 - 4x + 16 = 0$ using any method.

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by factoring, completing the square, or using the quadratic formula.

Key Terms and Formulas:

Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant: $b^2 - 4ac$

Step-by-Step Guidance

Identify $a = 1$, $b = -4$, $c = 16$.
Calculate the discriminant: $(-4)^2 - 4(1)(16)$.
Plug values into the quadratic formula.
Simplify under the square root and solve for $x$.

Try solving on your own before revealing the answer!

Final Answer:

No real solutions; discriminant is negative. The solutions are complex: $x = 2 \pm 2i$.

Q14. Circle which end behavior best describes the graph of $f(x) = x^4 + x^3 - x + 1$.

Background

Topic: End Behavior of Polynomial Functions

This question tests your understanding of how the degree and leading coefficient affect the graph's end behavior.

Key Terms:

Degree: Highest power of $x$.
Leading coefficient: Coefficient of the highest power.

Step-by-Step Guidance

Identify the degree ($4$) and leading coefficient ($1$).
For even degree and positive leading coefficient, both ends go up.
Describe the end behavior as $x \to \pm \infty$.

Try describing the end behavior before revealing the answer!

Final Answer:

As $x \to \pm \infty$, $f(x) \to \infty$. Both ends of the graph rise.

Q15. List all local and absolute extrema of the function defined by the graph. Write answers as $f(a) = b$. Write N/A if none.

Background

Topic: Extrema of Functions

This question tests your ability to identify local and absolute maxima and minima from a graph.

Key Terms:

Local maximum/minimum: Highest/lowest point in a neighborhood.
Absolute maximum/minimum: Highest/lowest point overall.

Step-by-Step Guidance

Examine the graph for peaks (maxima) and valleys (minima).
Identify local maxima/minima by checking points where the graph changes direction.
Compare all values to find absolute extrema.
Write answers in the form $f(a) = b$.

Try identifying extrema before revealing the answer!

Final Answer:

Local Maxima: $f(-0.693) = -0.397$ Local Minima: $f(1.443) = -2.833$ Absolute Maxima: $f(0) = 0$ Absolute Minima: $f(1.443) = -2.833$

Q16. Solve the polynomial equation $x^3 + x^2 - 4x - 4 = 0$.

Background

Topic: Solving Polynomial Equations

This question tests your ability to solve cubic equations, possibly by factoring or using the Rational Root Theorem.

Key Terms and Formulas:

Rational Root Theorem: Possible rational roots are factors of constant term over leading coefficient.
Factoring: Expressing the polynomial as a product of factors.

Step-by-Step Guidance

List possible rational roots: $\pm1, \pm2, \pm4$.
Test each possible root by substitution.
Once a root is found, use synthetic or long division to factor the cubic.
Solve the resulting quadratic equation.

Try factoring and solving before revealing the answer!

Final Answer:

Roots: $x = -1$, $x = -2$, $x = 2$

Q17. Sketch the graph of $f(x) = x^3 + x^2 - 4x - 4$. What is the end behavior, y-intercept, zeros, and behavior at zeros?

Background

Topic: Graphing Polynomial Functions

This question tests your ability to analyze and sketch a cubic polynomial.

Key Terms:

End behavior: How the graph behaves as $x \to \pm \infty$.
Y-intercept: Value at $x = 0$.
Zeros: Where $f(x) = 0$.

Step-by-Step Guidance

Determine end behavior: For cubic with positive leading coefficient, left end down, right end up.
Find y-intercept: Substitute $x = 0$.
Find zeros: Solve $x^3 + x^2 - 4x - 4 = 0$.
Analyze behavior at each zero (crosses or touches).
Sketch the graph using this information.

Try sketching and analyzing before revealing the answer!

Final Answer:

End behavior: As $x \to -\infty$, $f(x) \to -\infty$; as $x \to \infty$, $f(x) \to \infty$. Y-intercept: $f(0) = -4$ Zeros: $x = -2, -1, 2$ The graph crosses the x-axis at each zero.

Q18. Use the graph to solve the polynomial inequality $x^3 + x^2 - 4x - 4 \geq 0$.

Background

Topic: Polynomial Inequalities

This question tests your ability to use a graph to solve inequalities.

Key Terms:

Polynomial inequality: Find where the graph is above or on the x-axis.
Interval notation: Expressing solution as intervals.

Step-by-Step Guidance

Identify the zeros from the graph.
Determine where the graph is above or on the x-axis.
Write the solution in interval notation.

Try analyzing the graph before revealing the answer!

Final Answer:

Solution: $[-2, -1] \cup [2, \infty)$

Q19. Sketch the graph of $f(x) = \frac{-2}{x+2} + 3$. Give the domain and range.

Background

Topic: Rational Functions

This question tests your ability to graph rational functions and determine domain and range.

Key Terms and Formulas:

Vertical asymptote: $x = -2$
Horizontal asymptote: $y = 3$
Domain: All real numbers except where denominator is zero.
Range: All real numbers except horizontal asymptote.

Step-by-Step Guidance

Identify vertical asymptote: Set denominator to zero.
Identify horizontal asymptote: Consider end behavior as $x \to \pm \infty$.
Determine domain: Exclude $x = -2$.
Determine range: Exclude $y = 3$.
Sketch the graph showing asymptotes and general shape.

Try sketching and stating domain/range before revealing the answer!

Final Answer:

Domain: $(-\infty, -2) \cup (-2, \infty)$ Range: $(-\infty, 3) \cup (3, \infty)$ The graph has a vertical asymptote at $x = -2$ and horizontal asymptote at $y = 3$.

Q20. Let $f(x) = 3x - 4$ and $g(x) = 2x - 5$. Find $(fg)(x)$.

Background

Topic: Function Operations

This question tests your ability to multiply two functions.

Key Terms and Formulas:

Product of functions: $(fg)(x) = f(x) \cdot g(x)$

Step-by-Step Guidance

Write $f(x)$ and $g(x)$.
Multiply $f(x)$ and $g(x)$: $(3x - 4)(2x - 5)$.
Expand the product using distributive property.
Combine like terms.

Try expanding before revealing the answer!

Final Answer:

$(fg)(x) = 6x^2 - 23x + 20$

Q21. Let $f(x) = x^5 + 7$ and $g(x) = \sqrt{x - 2}$; find $(f \circ g)(x)$.

Background

Topic: Function Composition

This question tests your ability to compose two functions.

Key Terms and Formulas:

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

Step-by-Step Guidance

Write $g(x) = \sqrt{x - 2}$.
Substitute $g(x)$ into $f(x)$: $f(g(x)) = (\sqrt{x - 2})^5 + 7$.
Simplify the expression.

Try composing before revealing the answer!

Final Answer:

$(f \circ g)(x) = (\sqrt{x - 2})^5 + 7$

Q22. Find $(f - g)(-1)$ given the graphs of $f$ and $g$.

Background

Topic: Function Operations

This question tests your ability to subtract function values at a specific input.

Key Terms:

Function subtraction: $(f - g)(x) = f(x) - g(x)$

Step-by-Step Guidance

Find $f(-1)$ from the graph.
Find $g(-1)$ from the graph.
Subtract: $f(-1) - g(-1)$.

Try finding values before revealing the answer!

Final Answer:

$(f - g)(-1)$ is the difference between the values of $f$ and $g$ at $x = -1$ as shown on the graph.

Q23. Find $(f \circ g)(3)$ given the graphs of $f$ and $g$.

Background

Topic: Function Composition

This question tests your ability to compose functions using values from a graph.

Key Terms:

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

Step-by-Step Guidance

Find $g(3)$ from the graph.
Use $g(3)$ as input for $f$; find $f(g(3))$ from the graph.

Try finding values before revealing the answer!

Final Answer:

$(f \circ g)(3)$ is the value of $f$ at $g(3)$, as shown on the graph.

Q24. Let $f(x) = 2x - 7$ (one-to-one). Find $f^{-1}(x)$.

Background

Topic: Inverse Functions

This question tests your ability to find the inverse of a linear function.

Key Terms and Formulas:

Inverse function: $f^{-1}(x)$ reverses the effect of $f(x)$.

Step-by-Step Guidance

Write $y = 2x - 7$.
Swap $x$ and $y$: $x = 2y - 7$.
Solve for $y$.
Write $f^{-1}(x)$ in terms of $x$.

Try finding the inverse before revealing the answer!

Final Answer:

$f^{-1}(x) = \frac{x + 7}{2}$

Q25. Sketch the graph of the inverse of the given one-to-one function.

Background

Topic: Inverse Functions and Graphs

This question tests your ability to graph the inverse of a function.

Key Terms:

Inverse graph: Reflects the original graph over $y = x$.

Step-by-Step Guidance

Identify the original function's graph.
Reflect each point $(a, b)$ to $(b, a)$.
Draw the reflected graph.

Try sketching before revealing the answer!

Final Answer:

The graph of the inverse is a reflection of the original over the line $y = x$.

Q26. Sketch the graph and state the domain and range of $f(x) = 3x + 2 - 4$.

Background

Topic: Linear Functions

This question tests your ability to graph a linear function and determine domain and range.

Key Terms:

Linear function: $f(x) = mx + b$
Domain: All real numbers
Range: All real numbers

Step-by-Step Guidance

Simplify the function: $f(x) = 3x - 2$.
Identify slope and y-intercept.
Plot the y-intercept and use the slope to find another point.
Draw the line and state domain and range.

Try sketching and stating domain/range before revealing the answer!

Final Answer:

Domain: $(-\infty, \infty)$ Range: $(-\infty, \infty)$ The graph is a straight line with slope $3$ and y-intercept $-2$.

Q27. Suppose $2000 is invested at 6% annual interest compounded quarterly. How much money would you have in 9 years?

Background

Topic: Exponential Growth/Compound Interest

This question tests your ability to use the compound interest formula.

Key Terms and Formulas:

Compound interest formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$
$P$: Principal ($2000$)
$r$: Annual rate ($0.06$)
$n$: Number of compounding periods per year ($4$)
$t$: Number of years ($9$)

Step-by-Step Guidance

Identify all values: $P = 2000$, $r = 0.06$, $n = 4$, $t = 9$.
Plug values into the formula: $A = 2000\left(1 + \frac{0.06}{4}\right)^{4 \times 9}$.
Simplify inside the parentheses.
Calculate the exponent.
Multiply to find the final amount.

Try calculating before revealing the answer!

Final Answer:

$A \approx 3432.04$ After 9 years, the investment grows to about $3432.04$.

Q28. Find $\log_2 64$.

Background

Topic: Logarithms

This question tests your ability to evaluate logarithms.

Key Terms and Formulas:

Logarithm: $\log_b a$ is the exponent to which $b$ must be raised to get $a$.

Step-by-Step Guidance

Set $x = \log_2 64$.
Rewrite as $2^x = 64$.
Express $64$ as a power of $2$.
Find the exponent.

Try solving before revealing the answer!

Final Answer:

$\log_2 64 = 6$

Q29. Find $\log_5 411$ and round your answer to the nearest hundredth.

Background

Topic: Logarithms

This question tests your ability to evaluate logarithms using a calculator.

Key Terms and Formulas:

Change of base formula: $\log_b a = \frac{\log a}{\log b}$

Step-by-Step Guidance

Use the change of base formula: $\log_5 411 = \frac{\log 411}{\log 5}$.
Calculate $\log 411$ and $\log 5$ using a calculator.
Divide and round to the nearest hundredth.

Try calculating before revealing the answer!

Final Answer:

$\log_5 411 \approx 3.61$

Q30. Solve the exponential equation $98^x = 5x - 6$ and round the solution to three decimal places.

Background

Topic: Exponential Equations

This question tests your ability to solve equations involving exponentials, possibly using logarithms or graphing.

Key Terms and Formulas:

Exponential equation: An equation where the variable is in the exponent.
Logarithms: Used to solve for exponents.

Step-by-Step Guidance

Set $98^x = 5x - 6$.
Recognize that this equation may require graphing or numerical methods to solve.
Graph both sides or use logarithms to isolate $x$.
Round the solution to three decimal places.

Try solving before revealing the answer!

Final Answer:

$x \approx 0.073$