Functions, Domain, and Range – Intermediate Algebra Study Guide

Q1. Decide whether the following relation is a function. Then state the domain and range.

Background

Topic: Functions, Domain, and Range

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

Key Terms:

• Function: A relation in which each input (domain element) is paired with exactly one output (range element).

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

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

Step-by-Step Guidance

1. Examine the arrows in the diagram. Check if each element in the left set (domain) maps to only one element in the right set (range).

2. If any element in the domain maps to more than one element in the range, the relation is not a function.

3. List all the elements in the domain (the left set).

4. List all the elements in the range (the right set that receives arrows).

5. Check for repeated outputs or missing values in the range.

Try solving on your own before revealing the answer!

Final Answer:

The relation is a function because each input from the domain maps to exactly one output in the range.

Domain: {8, 6, 4, 2}

Range: {1, 3, 5}

Each element in the domain has only one arrow pointing to a single element in the range, so this satisfies the definition of a function.

Q2. Decide whether the following relation is a function. Then state the domain and range.

Background

Topic: Functions, Domain, and Range (Graphical Representation)

This question tests your ability to determine if a graph represents a function and to identify its domain and range.

Key Terms:

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Domain: All possible x-values for the graph.

• Range: All possible y-values for the graph.

Step-by-Step Guidance

1. Apply the vertical line test: Imagine drawing vertical lines across the graph. Does any line touch the circle at more than one point?

2. If so, the graph does not represent a function.

3. Identify the domain: Look at the x-values covered by the circle.

4. Identify the range: Look at the y-values covered by the circle.

5. Express the domain and range using interval notation.

Try solving on your own before revealing the answer!

Final Answer:

The relation is not a function because a vertical line can intersect the circle at two points.

Domain: All x-values between -3 and 3, so $[-3, 3]$

Range: All y-values between -3 and 3, so $[-3, 3]$

The circle fails the vertical line test, so it is not a function.

Q3. Decide whether the following relation is a function. Then state the domain and range.

Background

Topic: Functions, Domain, and Range (Piecewise Graph)

This question tests your ability to analyze a piecewise graph to determine if it represents a function and to find its domain and range.

Key Terms:

• Piecewise Function: A function defined by different expressions for different intervals of the domain.

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

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

• Range: All y-values the graph attains.

Step-by-Step Guidance

1. Apply the vertical line test: Check if any vertical line crosses the graph at more than one point.

2. If not, the graph represents a function.

3. Identify the domain: Look at the x-values where the graph is drawn.

4. Identify the range: Look at the y-values the graph reaches.

5. Express the domain and range using interval notation.

Try solving on your own before revealing the answer!

Final Answer:

The relation is a function because no vertical line crosses the graph more than once.

Domain: All real numbers (the graph extends left and right indefinitely), so $(-\infty, \infty)$

Range: All y-values from negative infinity up to the highest point on the graph, so $(-\infty, 6]$ (assuming the highest y-value is 6).

The graph passes the vertical line test, so it is a function.

Q4. Decide whether the following relation is a function. Then state the domain and range.

Background

Topic: Functions, Domain, and Range (Graphing)

This question likely asks you to graph a relation or function and then determine its domain and range. Since the grid is blank, you may need to plot points or draw a graph based on additional information.

Key Terms:

• Graph: Visual representation of a relation or function.

• Domain: All possible x-values.

• Range: All possible y-values.

Step-by-Step Guidance

1. Plot the points or graph as instructed in your worksheet.

2. Apply the vertical line test if you draw a graph.

3. Identify the domain by looking at the x-values covered by your graph.

4. Identify the range by looking at the y-values covered by your graph.

5. Express the domain and range using interval notation.

Try solving on your own before revealing the answer!

Final Answer:

The answer depends on what you graph. If you plot a function, use the vertical line test and list the domain and range accordingly.

If you graph a line, the domain and range are usually $(-\infty, \infty)$ unless restricted.

Q5. Decide whether the following relation is a function. Then state the domain and range.

Background

Topic: Functions, Domain, and Range (Graphing)

This question is similar to Q4 and likely asks you to graph a relation or function and then determine its domain and range. The grid is blank, so you may need to plot points or draw a graph based on additional information.

Key Terms:

• Graph: Visual representation of a relation or function.

• Domain: All possible x-values.

• Range: All possible y-values.

Step-by-Step Guidance

1. Plot the points or graph as instructed in your worksheet.

2. Apply the vertical line test if you draw a graph.

3. Identify the domain by looking at the x-values covered by your graph.

4. Identify the range by looking at the y-values covered by your graph.

5. Express the domain and range using interval notation.

Try solving on your own before revealing the answer!

Final Answer:

The answer depends on what you graph. If you plot a function, use the vertical line test and list the domain and range accordingly.

If you graph a line, the domain and range are usually $(-\infty, \infty)$ unless restricted.

Q12. The table of values below is from a linear function. What is the slope of the linear function?

Background

Topic: Slope of a Linear Function

This question tests your ability to calculate the slope from a table of values, which is a fundamental skill in algebra.

Key Formula:

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Step-by-Step Guidance

1. Pick any two points from the table. For example, $(x_1, y_1) = (1, 1.5)$ and $(x_2, y_2) = (3, 0.5)$.

2. Plug the values into the slope formula: $m = \frac{0.5 - 1.5}{3 - 1}$

3. Simplify the numerator and denominator separately.

4. Calculate the slope for these two points.

5. Check with another pair of points to confirm the slope is consistent (since the function is linear).

Try solving on your own before revealing the answer!

Final Answer:

The slope of the linear function is $m = -0.5$.

This value is consistent for any pair of points in the table, confirming the function is linear.