Use the graph of the function to find the domain and range of f. (Enter your answers using interval notation.) y = f() domain range Use the graph to find the indicated function values. (a) f(-1) X (b) f(0) (c) f(1) : (d) f(2)
The Correct Answer and Explanation is:
Let us carefully analyze the graph provided and answer each part accordingly.
Domain and Range
The domain represents all the x-values for which the function exists. From the graph, the function starts at x=−4x = -4x=−4 and goes up to x=4x = 4x=4. At x=−4x = -4x=−4, the point is open, meaning it is not included in the domain. At x=4x = 4x=4, the point is closed, meaning it is included in the domain. Therefore, the domain in interval notation is: (−4,4](-4, 4](−4,4]
The range represents all the y-values that the function takes. Observing the graph vertically, the function has its lowest point around y=−4y = -4y=−4 and goes up to y=6y = 6y=6. Both ends appear to be closed at y=−4y = -4y=−4 and y=6y = 6y=6. Thus, the range in interval notation is: [−4,6][-4, 6][−4,6]
Function Values
Next, we determine the exact function values at the given points:
(a) f(−1)f(-1)f(−1): From the graph, at x=−1x = -1x=−1, the corresponding y-value is y=−4y = -4y=−4.
Thus, f(−1)=−4f(-1) = -4f(−1)=−4.
(b) f(0)f(0)f(0): At x=0x = 0x=0, the corresponding y-value is y=−3y = -3y=−3.
Thus, f(0)=−3f(0) = -3f(0)=−3.
(c) f(1)f(1)f(1): At x=1x = 1x=1, the corresponding y-value is y=−2y = -2y=−2.
Thus, f(1)=−2f(1) = -2f(1)=−2.
(d) f(2)f(2)f(2): At x=2x = 2x=2, the corresponding y-value is y=0y = 0y=0.
Thus, f(2)=0f(2) = 0f(2)=0.
Summary of Answers
- Domain: (−4,4](-4, 4](−4,4]
- Range: [−4,6][-4, 6][−4,6]
- f(−1)=−4f(-1) = -4f(−1)=−4
- f(0)=−3f(0) = -3f(0)=−3
- f(1)=−2f(1) = -2f(1)=−2
- f(2)=0f(2) = 0f(2)=0
Explanation
By examining the graph visually, each x-value has been traced directly to its corresponding y-value. The domain and range are found by analyzing the extent of the graph horizontally and vertically. Open and closed circles play a crucial role in determining inclusivity in the interval notation. This step-by-step analysis helps avoid confusion and ensures correct interpretation of the graph features.
