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: Determine the domain of the function by looking at the x-values over which the graph is defined. Since the graph extends indefinitely to the left and right, the domain is all real numbers, which can be written as \((-\infty, \infty)\).
2. Step 2: Determine the range of the function by identifying the minimum and maximum y-values the graph attains. From the graph, the highest point is at y = 4 and the lowest point is at y = -4, so the range is \([-4, 4]\).
3. Step 3: Identify the x-intercepts by finding the points where the graph crosses the x-axis (where y = 0). The graph crosses the x-axis at \(x = -11\), \(x = 0\), and \(x = 15\), so the x-intercepts are \((-11, 0)\), \((0, 0)\), and \((15, 0)\).
4. Step 4: Identify the y-intercept by finding the point where the graph crosses the y-axis (where x = 0). The graph crosses the y-axis at \((0, 0)\), so the y-intercept is \((0, 0)\).
5. Step 5: For the missing function values indicated by question marks, use the graph to read the corresponding y-values at the given x-values. Locate each x-value on the x-axis, then find the point on the graph directly above or below it to determine the function value \(f(x)\).

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 a graph, the domain corresponds to the horizontal extent of the curve. For example, if the graph extends infinitely left and right, the domain is all real numbers.

Range of a Function

The range of a function is the set of all possible output values (y-values) the function can produce. On a graph, the range is the vertical span covered by the curve. Identifying the highest and lowest points helps determine the range.

Intercepts of a Function

Intercepts are points where the graph crosses the axes. The x-intercepts occur where the function's value is zero (y=0), indicating roots of the function. The y-intercept is where the graph crosses the y-axis (x=0), showing the function's value at zero input.