In Exercises 77–92, 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; and e. the missing function values, indicated by question marks, below each graph.

Verified step by step guidance

1. Step 1: Identify the domain of the function by looking at the x-values for which the graph is defined. Notice the vertical asymptote at x = -3, which means the function is not defined at x = -3. Therefore, the domain includes all real numbers except x = -3.
2. Step 2: Determine the range of the function by observing the y-values that the graph takes. Notice the horizontal asymptote at y = 1, which the graph approaches but never reaches. The function values cover all y-values except y = 1.
3. Step 3: Find the x-intercepts by locating points where the graph crosses the x-axis (where y = 0). Check if the graph touches or crosses the x-axis at any point.
4. Step 4: Find the y-intercept by identifying the point where the graph crosses the y-axis (where x = 0). Read the y-value of the graph at x = 0.
5. Step 5: Use the graph to find the missing function values indicated by question marks by locating the corresponding x-values on the graph and reading off the y-values.

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the graph, vertical asymptotes indicate values excluded from the domain because the function approaches infinity or is undefined there. For example, the vertical dashed line at x = -4 shows the function is not defined at that point.

Range of a Function

The range is the set of all possible output values (y-values) that the function can take. Horizontal asymptotes, like the line y = 0 in the graph, indicate values that the function approaches but never reaches, helping to determine the range limits. Observing the graph's behavior helps identify the range.

Intercepts of a Function

Intercepts are points where the graph crosses the axes. The x-intercepts occur where the function equals zero (y=0), and the y-intercept is where the function crosses the y-axis (x=0). Identifying these points on the graph helps understand the function's behavior and solve for specific values.