Domain, all real numbers, ( - ∞,∞ )
Range, all y ≥ - 3, [ - 3,∞ )
Range, all y ≥ - 3, [ - 3,∞ )
To determine the range of the absolute value function, we need to identify the set of all possible output values (y-values) for the function. Since the graph opens up, the minimum value of the function is the y-coordinate of the vertex, which is -3. As the absolute value function continues to increase, there is no maximum value. Therefore, the range of the function is (-3, ∞).
The general form of an absolute value function is f(x) = |x - h| + k, where (h, k) represents the vertex of the function.
In this case, we are given that the vertex is at (0, -3). Therefore, our function can be written as f(x) = |x - 0| -3 = |x| - 3.
The vertex of the absolute value function is the lowest point on the graph if it opens upward or the highest point if it opens downward. Since the given absolute value function opens upward, the vertex (0, -3) is the lowest point on the graph.
Now, let's determine the domain of the function. The domain of an absolute value function is the set of all possible x-values for which the function is defined. In other words, it is the set of all real numbers.
Therefore, the domain of the given function is (-∞, ∞).
Next, let's determine the range of the function. The range of an absolute value function is the set of all possible y-values or the output values.
Since the graph of this absolute value function opens upward, it is symmetric about the vertex. This means that there is no upper bound on the y-values, and the minimum y-value is the y-coordinate of the vertex, which is -3.
Therefore, the range of the given function is (-3, ∞).