Q1. Find the x-intercept for the graph given below. Also, state the domain of the function in set-builder notation.

Background

Topic: Graphs of Functions, Intercepts, and Domain

This question tests your ability to read a graph, identify the x-intercept (where the graph crosses the x-axis), and determine the domain of the function using set-builder notation.

Key Terms and Formulas

• x-intercept: The value(s) of x where the graph crosses the x-axis (i.e., where ).

• Domain: The set of all possible x-values for which the function is defined.

• Set-builder notation: A way to describe a set by stating the properties that its members must satisfy, e.g., .

Step-by-Step Guidance

1. Examine the graph and identify the point(s) where the red curve crosses the x-axis. These are your x-intercepts.

2. To find the x-intercept, look for the value of x where (i.e., where the function value is zero).

3. Next, determine the domain by observing the leftmost and rightmost x-values for which the graph exists. Pay attention to whether the endpoints are included (closed circle or solid line) or excluded (open circle or gap).

4. Express the domain in set-builder notation, using inequalities to describe the interval of x-values where the function is defined.

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Q2. Based on the given graph, for what value of x is ?

Background

Topic: Reading Function Values from a Graph

This question asks you to find the x-value where the function's output is 5, i.e., where the graph reaches a height of 5 on the y-axis.

Key Terms and Formulas

• : This means you are looking for the x-value(s) where the y-coordinate is 5.

Step-by-Step Guidance

1. Locate the horizontal line 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.

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Q3. Find the value of at by using the graph shown.

Background

Topic: Evaluating Functions Using Graphs

This question tests your ability to use a graph to find the value of a function at a specific input.

Key Terms and Formulas

• : The output of the function for a given input .

Step-by-Step Guidance

1. Find on the x-axis of the graph.

2. Move vertically from until you reach the graph of the function.

3. Read the y-coordinate at this point. This is the value of .

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Q4. Determine the domain and the range using the graph shown.

Background

Topic: Domain and Range from Graphs

This question asks you to identify the set of all possible input values (domain) and output values (range) for a function based on its graph.

Key Terms and Formulas

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

• Range: All y-values that the function can take.

Step-by-Step Guidance

1. Look at the leftmost and rightmost points of the graph to determine the domain (x-values covered by the graph).

2. Look at the lowest and highest points of the graph to determine the range (y-values covered by the graph).

3. Check if the endpoints are included (solid dots or lines) or excluded (open circles or gaps) to decide on using brackets [ ] or parentheses ( ).

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Q5. Plot the following point in the Cartesian coordinate system:

Background

Topic: Plotting Points in the Coordinate Plane

This question tests your understanding of how to plot a point given in fractional coordinates on the Cartesian plane.

Key Terms and Formulas

• Cartesian coordinate system: A plane defined by an x-axis (horizontal) and a y-axis (vertical).

• Point : is the horizontal position, is the vertical position.

Step-by-Step Guidance

1. Convert the fractions to decimals if it helps: , .

2. On the graph, start at the origin (0,0). Move left to .

3. From there, move up to .

4. Mark the point at this location.

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Q6. Determine the x and y-intercepts (if there is any) of the graph shown.

Background

Topic: Intercepts of Graphs

This question asks you to find where the graph crosses the x-axis (x-intercepts) and y-axis (y-intercept).

Key Terms and Formulas

• x-intercept: Where .

• y-intercept: Where .

Step-by-Step Guidance

1. Look for points where the graph crosses the x-axis. These are your x-intercepts.

2. Look for the point where the graph crosses the y-axis. This is your y-intercept.

3. Record the x-values for x-intercepts and the y-value for the y-intercept.

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