College Algebra: Basics of Functions and Graphs – Step-by-Step Study Guidance

Q1. Determine whether the relation is a function. Give the domain and the range of the relation. {(4, 3), (4, 7), (7, 3), (7, 7)}

Background

Topic: Functions and Relations

This question tests your understanding of what makes a relation a function, and how to identify the domain and range from a set of ordered pairs.

Key Terms and Concepts:

• Function: A relation where each input (x-value) is paired with exactly one output (y-value).

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

Step-by-Step Guidance

1. List all the x-values (inputs) and y-values (outputs) from the given set of ordered pairs.

2. Check if any x-value is paired with more than one y-value. If so, the relation is not a function.

3. Write the domain as the set of all unique x-values.

4. Write the range as the set of all unique y-values.

Try solving on your own before revealing the answer!

Final Answer:

No, this relation is not a function because the input 4 is paired with both 3 and 7, and the input 7 is also paired with both 3 and 7.

The domain is {4, 7}.

The range is {3, 7}.

Q2. Determine whether the following relation represents a function. If it is a function, state the domain and range. {(−3, 6), (−1, 0), (2, 2), (0, 0)}

Background

Topic: Functions and Relations

This question checks your ability to determine if a set of ordered pairs forms a function and to identify the domain and range.

Key Terms and Concepts:

• Function: Each x-value must be paired with only one y-value.

• Domain: All unique x-values.

• Range: All unique y-values.

Step-by-Step Guidance

1. List all x-values and check if any x-value repeats with a different y-value.

2. If no x-value repeats, the relation is a function.

3. List the domain by collecting all unique x-values.

4. List the range by collecting all unique y-values.

Try solving on your own before revealing the answer!

Final Answer:

Yes, this relation is a function because each x-value is paired with only one y-value.

The domain is {−3, −1, 2, 0}.

The range is {6, 0, 2}.

Q3. Use the graph to determine the function's domain and range.

Background

Topic: Domain and Range from Graphs

This question tests your ability to read a graph and identify the set of all possible x-values (domain) and y-values (range) for a function.

Key Terms and Concepts:

• Domain: The set of all x-values for which the function is defined (where the graph exists horizontally).

• Range: The set of all y-values the function takes (where the graph exists vertically).

Step-by-Step Guidance

1. Look at the graph and identify the leftmost and rightmost points to determine the domain.

2. Check if the graph continues indefinitely or stops at certain x-values.

3. Identify the lowest and highest points on the graph to determine the range.

4. Check if the graph continues up or down without bound, or if it stops at certain y-values.

Try solving on your own before revealing the answer!

Final Answer:

The domain is [0, ∞).

The range is [−3, ∞).

Q4. Determine if the following statement is true or false: The relation $y = \pm \sqrt{x^2 - 6}$ defines y as a function of x.

Background

Topic: Functions Defined by Equations

This question tests your understanding of when an equation defines y as a function of x, especially when there is a plus-minus (±) symbol.

Key Terms and Concepts:

• Function: For each x in the domain, there must be exactly one y-value.

• Plus-minus (±): Indicates two possible values for y for each x.

Step-by-Step Guidance

1. Consider what happens when you plug in a value for x. Does each x give only one y-value, or more than one?

2. Think about the meaning of the ± symbol in the equation.

3. Recall the definition of a function: one input, one output.

Try solving on your own before revealing the answer!

Final Answer:

False. The equation $y = \pm \sqrt{x^2 - 6}$ does not define y as a function of x because for each x (where $x^2 - 6 \geq 0$), there are two possible y-values (one positive and one negative).

Q5. Determine whether the following equation defines y as a function of x: $x + y = 49$

Background

Topic: Functions Defined by Equations

This question asks you to solve for y in terms of x and determine if each x gives exactly one y-value.

Key Terms and Concepts:

• Function: Each x-value must correspond to exactly one y-value.

Step-by-Step Guidance

1. Rearrange the equation to solve for y in terms of x.

2. Check if, for each x, there is only one corresponding y-value.

3. Consider if there are any restrictions or multiple solutions for y for a given x.

Try solving on your own before revealing the answer!

Final Answer:

Yes, $x + y = 49$ defines y as a function of x because for each x, there is exactly one y-value: $y = 49 - x$.

Q6. Determine whether the following equation defines y as a function of x: $36 = x + y^2$

Background

Topic: Functions Defined by Equations

This question tests your ability to solve for y and determine if the equation defines y as a function of x.

Key Terms and Concepts:

• Function: Each x-value must correspond to exactly one y-value.

• Quadratic Equation: Solving for y may involve taking a square root, which can yield two values.

Step-by-Step Guidance

1. Rearrange the equation to solve for y in terms of x.

2. When you isolate y, check if you need to take a square root.

3. Consider whether each x-value gives one or more y-values.

Try solving on your own before revealing the answer!

Final Answer:

No, $36 = x + y^2$ does not define y as a function of x because solving for y gives $y = \pm \sqrt{36 - x}$, which means two possible y-values for each x (where $36 - x \geq 0$).

Q7. Determine whether the following equation defines y as a function of x: $36 = (x + y)^2$

Background

Topic: Functions Defined by Equations

This question asks you to solve for y and determine if each x gives only one y-value.

Key Terms and Concepts:

• Function: Each x-value must correspond to exactly one y-value.

• Square Root: Taking a square root can yield two solutions (positive and negative).

Step-by-Step Guidance

1. Rewrite the equation to isolate (x + y).

2. Take the square root of both sides, remembering to consider both the positive and negative roots.

3. Solve for y in terms of x and check if each x gives one or more y-values.

Try solving on your own before revealing the answer!

Final Answer:

No, $36 = (x + y)^2$ does not define y as a function of x because solving for y gives two possible values for each x: $y = -x + 6$ and $y = -x - 6$.

Q8. Evaluate the function at the given values and simplify: $f(x) = 6x + 7$ a) $f(3)$ b) $f(x + 4)$ c) $f(-x)$

Background

Topic: Evaluating Functions

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

Key Terms and Concepts:

• Function Evaluation: Substitute the given value or expression for x in the function.

Step-by-Step Guidance

1. For part (a), substitute 3 for x in $f(x)$ and simplify.

2. For part (b), substitute $(x + 4)$ for x in $f(x)$ and simplify.

3. For part (c), substitute $-x$ for x in $f(x)$ and simplify.

Try solving on your own before revealing the answer!

Final Answer:

a) $f(3) = 6 \times 3 + 7 = 18 + 7 = 25$

b) $f(x + 4) = 6(x + 4) + 7 = 6x + 24 + 7 = 6x + 31$

c) $f(-x) = 6(-x) + 7 = -6x + 7$

Q9. Evaluate the function at the given values and simplify: $f(x) = x^2 + 3x + 2$ a) $f(7)$ b) $f(x + 9)$ c) $f(-x)$

Background

Topic: Evaluating Functions

This question tests your ability to substitute values and expressions into a quadratic function and simplify.

Key Terms and Concepts:

• Function Evaluation: Substitute the given value or expression for x in the function and simplify using algebraic rules.

Step-by-Step Guidance

1. For part (a), substitute 7 for x in $f(x)$ and simplify.

2. For part (b), substitute $(x + 9)$ for x in $f(x)$ and expand and simplify.

3. For part (c), substitute $-x$ for x in $f(x)$ and simplify.

Try solving on your own before revealing the answer!

Final Answer:

a) $f(7) = 7^2 + 3 \times 7 + 2 = 49 + 21 + 2 = 72$

b) $f(x + 9) = (x + 9)^2 + 3(x + 9) + 2 = x^2 + 18x + 81 + 3x + 27 + 2 = x^2 + 21x + 110$

c) $f(-x) = (-x)^2 + 3(-x) + 2 = x^2 - 3x + 2$

Q10. Evaluate the function at the given values and simplify: $f(x) = \frac{6x^2 - 1}{x^2}$ a) $f(3)$ b) $f(-3)$ c) $f(-x)$

Background

Topic: Evaluating Rational Functions

This question tests your ability to substitute values and expressions into a rational function and simplify.

Key Terms and Concepts:

• Rational Function: A function that is the ratio of two polynomials.

Step-by-Step Guidance

1. For part (a), substitute 3 for x in $f(x)$ and simplify the numerator and denominator.

2. For part (b), substitute $-3$ for x in $f(x)$ and simplify.

3. For part (c), substitute $-x$ for x in $f(x)$ and simplify.

Try solving on your own before revealing the answer!

Final Answer:

a) $f(3) = \frac{6 \times 9 - 1}{9} = \frac{54 - 1}{9} = \frac{53}{9}$

b) $f(-3) = \frac{6 \times 9 - 1}{9} = \frac{53}{9}$

c) $f(-x) = \frac{6(-x)^2 - 1}{(-x)^2} = \frac{6x^2 - 1}{x^2}$

Q11. Use the vertical line test to determine if y is a function of x in the graph.

Background

Topic: Vertical Line Test

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

Key Terms and Concepts:

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

Step-by-Step Guidance

1. Imagine drawing vertical lines at various x-values across the graph.

2. Check if any vertical line touches the graph at more than one point.

3. If all vertical lines touch the graph at most once, it is a function.

Try solving on your own before revealing the answer!

Final Answer:

If every vertical line crosses the graph at most once, then y is a function of x. Otherwise, it is not.

Q12. Use the vertical line test to determine if y is a function of x.

Background

Topic: Vertical Line Test

This question is similar to the previous one and tests your understanding of the vertical line test for functions.

Key Terms and Concepts:

• Vertical Line Test: Used to determine if a graph represents a function.

Step-by-Step Guidance

1. Visualize or draw vertical lines at different x-values on the graph.

2. Observe if any vertical line crosses the graph more than once.

3. Decide if the graph passes or fails the vertical line test.

Try solving on your own before revealing the answer!

Final Answer:

If any vertical line crosses the graph more than once, y is not a function of x. If not, it is a function.

Q13. Use the vertical line test to determine if y is a function of x in the graph.

Background

Topic: Vertical Line Test

This question again asks you to apply the vertical line test to a graph to determine if it represents a function.

Key Terms and Concepts:

• Vertical Line Test: A method to check if a graph is a function.

Step-by-Step Guidance

1. Draw or imagine vertical lines at various x-values.

2. Check if any vertical line crosses the graph more than once.

3. Use your observation to decide if the graph represents a function.

Try solving on your own before revealing the answer!

Final Answer:

If every vertical line crosses the graph at most once, y is a function of x. Otherwise, it is not.

Q14. Use the graph of f to find the value of f(1).

Background

Topic: Evaluating Functions from Graphs

This question tests your ability to read a graph and find the value of the function at a specific x-value.

Key Terms and Concepts:

• Function Value: The y-value on the graph corresponding to a given x-value.

Step-by-Step Guidance

1. Locate x = 1 on the x-axis of the graph.

2. Move vertically to the point on the graph above or below x = 1.

3. Read the corresponding y-value at that point; this is f(1).

Try solving on your own before revealing the answer!

Final Answer:

f(1) = 0 (the y-value at x = 1 on the graph is 0).

Q15. Use the graph of g to find g(−8).

Background

Topic: Evaluating Functions from Graphs

This question tests your ability to find the value of a function at a specific x-value using its graph.

Key Terms and Concepts:

• Function Value: The y-value corresponding to a given x-value on the graph.

Step-by-Step Guidance

1. Find x = −8 on the x-axis of the graph.

2. Move vertically to the graph at x = −8.

3. Read the y-value at that point; this is g(−8).

Try solving on your own before revealing the answer!

Final Answer:

g(−8) = 9 (the y-value at x = −8 on the graph is 9).

Q16. For what value of x is g(x) = 4?

Background

Topic: Solving Equations Using Graphs

This question tests your ability to use a graph to find the x-value(s) where the function equals a specific y-value.

Key Terms and Concepts:

• Solving for x: Find the x-value where the graph crosses the horizontal line y = 4.

Step-by-Step Guidance

1. Draw or imagine a horizontal line at y = 4 on the graph.

2. Find the point(s) where the graph intersects this line.

3. Read the corresponding x-value(s) at these intersection points.

Try solving on your own before revealing the answer!

Final Answer:

x = −4 (the graph of g(x) crosses y = 4 at x = −4).

Q17. Use the graph to determine: a) the function's domain b) the function's range c) the x-intercepts, if any d) the y-intercept, if any e) the missing function values: f(−2) and f(2)

Background

Topic: Interpreting Graphs of Functions

This question tests your ability to extract domain, range, intercepts, and specific function values from a graph.

Key Terms and Concepts:

• Domain: All x-values for which the graph exists.

• Range: All y-values the graph attains.

• x-intercept: Where the graph crosses the x-axis (y = 0).

• y-intercept: Where the graph crosses the y-axis (x = 0).

• Function Value: The y-value for a given x-value.

Step-by-Step Guidance

1. For the domain, look at the leftmost and rightmost x-values where the graph exists.

2. For the range, find the lowest and highest y-values on the graph.

3. For x-intercepts, find where the graph crosses the x-axis.

4. For the y-intercept, find where the graph crosses the y-axis.

5. For f(−2) and f(2), find the y-values at x = −2 and x = 2.

Try solving on your own before revealing the answer!

Final Answer:

a) Domain: (−∞, ∞)

b) Range: (−∞, 1]

c) x-intercepts: −1, 1

d) y-intercept: 1

e) f(−2) = −3, f(2) = −3

Q18. Use the graph to determine: a) the function's domain b) the function's range c) the x-intercepts, if any d) the y-intercept, if any e) the missing function value: f(4)

Background

Topic: Interpreting Graphs of Functions

This question tests your ability to read domain, range, intercepts, and specific values from a graph.

Key Terms and Concepts:

• Domain: All x-values for which the graph exists.

• Range: All y-values the graph attains.

• x-intercept: Where the graph crosses the x-axis (y = 0).

• y-intercept: Where the graph crosses the y-axis (x = 0).

• Function Value: The y-value for a given x-value.

Step-by-Step Guidance

1. For the domain, look at the interval of x-values where the graph exists.

2. For the range, find the interval of y-values the graph covers.

3. For intercepts, check where the graph crosses the axes.

4. For f(4), find the y-value at x = 4.

Try solving on your own before revealing the answer!

Final Answer:

a) Domain: [0, ∞)

b) Range: (−∞, −2]

c) There is no x-intercept.

d) y-intercept: −2

e) f(4) = −4

Q19. Use the graph to determine: a) the function's domain b) the function's range c) the x-intercepts, if any d) the y-intercept, if any e) the missing function value: f(1)

Background

Topic: Interpreting Graphs of Functions

This question tests your ability to extract domain, range, intercepts, and specific function values from a graph.

Key Terms and Concepts:

• Domain: All x-values for which the graph exists.

• Range: All y-values the graph attains.

• x-intercept: Where the graph crosses the x-axis (y = 0).

• y-intercept: Where the graph crosses the y-axis (x = 0).

• Function Value: The y-value for a given x-value.

Step-by-Step Guidance

1. For the domain, look at the interval of x-values where the graph exists.

2. For the range, find the interval of y-values the graph covers.

3. For intercepts, check where the graph crosses the axes.

4. For f(1), find the y-value at x = 1.

Try solving on your own before revealing the answer!

Final Answer:

a) Domain: (−∞, ∞)

b) Range: (−∞, 0)

c) There is no x-intercept.

d) y-intercept: −1

e) f(1) = −2

Q20. The bar graph shows the wage gap between men and women for selected years from 1980 through 2010. The function $G(x) = -0.01x^2 + x + 55$ models the wage gap, as a percent, x years after 1980. Use this information to complete parts a and b. a) Find and interpret $G(30)$. b) Does $G(30)$ overestimate or underestimate the actual data shown by the bar graph? By how much?

Background

Topic: Function Modeling and Interpretation

This question tests your ability to evaluate a quadratic function and interpret its meaning in a real-world context, as well as compare model predictions to actual data.

Key Terms and Concepts:

• Function Evaluation: Substitute the given value for x in the function and simplify.

• Interpretation: Understand what the function value means in the context of the problem.

• Comparison: Compare the function's output to actual data to determine if it overestimates or underestimates.

Step-by-Step Guidance

1. For part (a), substitute x = 30 into $G(x)$ and simplify to find $G(30)$.

2. Interpret what $G(30)$ means in terms of the wage gap and the year (2010).

3. For part (b), compare $G(30)$ to the actual wage gap value from the bar graph for 2010.

4. Calculate the difference between the model's prediction and the actual value to determine if it is an overestimate or underestimate.

Try solving on your own before revealing the answer!

Final Answer:

a) $G(30) = -0.01(30)^2 + 30 + 55 = -0.01(900) + 30 + 55 = -9 + 30 + 55 = 76$

This represents a wage gap of 76% in the year 2010 (30 years after 1980).

b) The actual wage gap in 2010 is 81%. The function underestimates the actual data by 5% (81% − 76% = 5%).