Use the function below to answer the following parts.
1/3x - 4 if -6 </= x < - 3
f(x) = 2 if -3 < x < 3
x^2 - 4 if x > 3
I found out the domain, range, and points of disontinuity. I think i did them right:
Domain= [-6,infinity]
Range= [-6,infinity]
and points of discontinuity = x=-3,x=3
How do i graph these functions?
Your first segment runs from (-6,-6) to (-3,-5)
for -6 ≤ x < -3
so draw a solid dot for (-6,-6) and an "open" dot for (-3,-5)
the 2nd segment runs from (-3,2) to (3,2).
draw that line leaving the end points as open dots.
the 3rd graph is part of a parabola, which has its vertex at (0,-4) and which would open upwards.
I would pass through (1,-3), (2,0) and (3,5)
draw it only from (3,5) on, leaving the (3,5) open.
you are correct with your domain and range, and your value of x that have a discontinuity.
thanks
To graph the function, you will need to plot points on a coordinate plane based on the given function. Here are the steps to graph each part of the function:
1) For the interval -6 ≤ x < -3:
- Choose some values within this range, such as x = -6, -5, -4, -3. Plug these values into the function and calculate the corresponding y-values:
f(-6) = (1/3)(-6) - 4 = -2
f(-5) = (1/3)(-5) - 4 = -4.67
f(-4) = (1/3)(-4) - 4 = -4.33
f(-3) = (1/3)(-3) - 4 = -4
- Plot the points (-6, -2), (-5, -4.67), (-4, -4.33), and (-3, -4). Connect these points with a straight line segment.
2) For the interval -3 < x < 3:
- Within this interval, the function is constant and equal to 2. Therefore, you can simply draw a horizontal line at y = 2 throughout this interval.
3) For x > 3:
- Choose some values within this range, such as x = 3, 4, 5, 6. Plug these values into the function and calculate the corresponding y-values:
f(3) = 3^2 - 4 = 5
f(4) = 4^2 - 4 = 12
f(5) = 5^2 - 4 = 21
f(6) = 6^2 - 4 = 32
- Plot the points (3, 5), (4, 12), (5, 21), and (6, 32). Connect these points with a smooth curve.
Remember, there are points of discontinuity at x = -3 and x = 3, so you should draw open circles at those points on the graph, indicating that the function is not defined at those specific values.
Lastly, note that the domain and range you provided are incorrect. The correct domain and range for this function are:
- Domain: (-∞, -3] ∪ (-3, 3) ∪ [3, ∞)
- Range: [-4, ∞)