College Algebra Test 2 Review – Step-by-Step Guidance

Q13. Explain what the vertical line test is for and how to use it.

Background

Topic: Functions and Graphs

This question tests your understanding of how to determine if a graph represents a function using the vertical line test.

Key Terms:

• Function: A relation in which each input (x-value) has exactly one output (y-value).

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

Step-by-Step Guidance

1. Imagine drawing vertical lines (parallel to the y-axis) across the entire graph.

2. Observe how many times each vertical line intersects the graph.

3. If any vertical line crosses the graph more than once, the graph does not represent a function.

4. If every vertical line crosses the graph at most once, the graph does represent a function.

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Q14. For each relation, state if it is a function. State the domain and the range.

Background

Topic: Relations and Functions

This question asks you to determine if a set of ordered pairs is a function, and to identify the domain and range.

Key Terms:

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

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

• Function: Each input has exactly one output.

Step-by-Step Guidance

1. For each relation, list all the x-values (domain) and y-values (range).

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

3. If each x-value is paired with only one y-value, it is a function.

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Q18. The graph of the function g is shown here.

Background

Topic: Graph Analysis – Increasing/Decreasing Intervals, Even/Odd Functions

This question asks you to analyze a graph to determine where the function is increasing, decreasing, or constant, and whether it is even, odd, or neither.

Key Terms:

• Increasing Interval: Where the graph rises as you move left to right.

• Decreasing Interval: Where the graph falls as you move left to right.

• Constant Interval: Where the graph is flat (horizontal).

• Even Function: Symmetric about the y-axis.

• Odd Function: Symmetric about the origin.

Step-by-Step Guidance

1. Look at the graph and identify the intervals where the function is going up (increasing) or down (decreasing).

2. Check for any flat (horizontal) sections to determine constant intervals.

3. Examine the graph for symmetry about the y-axis (even) or origin (odd).

4. To evaluate $g(3)$ and $g(-3)$, find the y-values at $x = 3$ and $x = -3$ on the graph.

Try solving on your own before revealing the answer!