Q1a. Find the domain and range of the function from its graph (Graph G(x)).

Background

Topic: Domain and Range from Graphs

This question tests your ability to determine the set of all possible input values (domain) and output values (range) for a function by analyzing its graph. You will need to look at the graph from left to right for the domain and from bottom to top for the range.

Key Terms and Concepts:

• Domain: All x-values for which the function is defined (where the graph exists horizontally).

• Range: All y-values that the function attains (where the graph exists vertically).

• Open Circle (○): Endpoint not included in the domain or range.

• Closed Circle (●): Endpoint included in the domain or range.

• Arrow: The graph continues indefinitely in that direction.

Step-by-Step Guidance

1. Look at the graph from left to right to determine the smallest and largest x-values that the graph reaches. Note if the endpoints are open or closed circles, or if there are arrows indicating the graph continues forever.

2. Write the domain in interval notation, using brackets [ ] for included endpoints and parentheses ( ) for excluded endpoints.

3. Now, look at the graph from bottom to top to determine the smallest and largest y-values that the graph reaches. Again, check for open/closed circles or arrows.

4. Write the range in interval notation, using the same rules for brackets and parentheses.

5. Check your intervals to make sure they match what you see on the graph.

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Q1b. Find the domain and range of the function from its graph (Graph h(z)).

Background

Topic: Domain and Range from Graphs

This question also asks you to determine the domain and range by analyzing the graph of a function. The process is the same as in part (a).

Key Terms and Concepts:

• Domain: All possible x-values (z-values in this case) for which the function is defined.

• Range: All possible y-values the function attains.

• Check for arrows, open/closed circles.

Step-by-Step Guidance

1. Scan the graph from left to right to find the smallest and largest z-values (domain).

2. Note if the graph continues indefinitely (arrows) or stops at endpoints (circles).

3. Write the domain in interval notation, using brackets or parentheses as appropriate.

4. Scan the graph from bottom to top to find the smallest and largest y-values (range).

5. Write the range in interval notation, using the correct symbols for included/excluded values.

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Q1c. Find the domain and range of the function from its graph (Graph f(t)).

Background

Topic: Domain and Range from Graphs

This question continues to test your ability to read domain and range from a graph, with attention to open and closed circles.

Key Terms and Concepts:

• Domain: All t-values for which the function is defined.

• Range: All y-values the function attains.

• Open circle: Value not included; Closed circle: Value included.

Step-by-Step Guidance

1. Look at the graph from left to right to determine the smallest and largest t-values (domain). Pay attention to open and closed circles at the endpoints.

2. Write the domain in interval notation, using ( ) for open circles and [ ] for closed circles.

3. Look at the graph from bottom to top to determine the smallest and largest y-values (range). Again, check for open/closed circles.

4. Write the range in interval notation, using the correct symbols for included/excluded values.

5. Double-check your intervals to match the graph's endpoints.

Try solving on your own before revealing the answer!