Drow the graph of power function y=x⅔
and show the table values
Domain =?
Range=?
To plot the graph of the power function y = x^(2/3), we can choose specific values for x and calculate the corresponding values for y. Let's choose x-values from -10 to 10.
x | y = x^(2/3)
--------------------
-10 | 10^(2/3) = 10^(2/3) ≈ 4.6416
-5 | (-5)^(2/3) ≈ 2.924
-3 | (-3)^(2/3) ≈ 2.0801
-1 | (-1)^(2/3) = (-1)^(2/3) = 1
0 | 0^(2/3) = 0^(2/3) = 0
1 | 1^(2/3) = 1^(2/3) = 1
3 | 3^(2/3) ≈ 1.8171
5 | 5^(2/3) ≈ 2.924
10 | 10^(2/3) ≈ 4.6416
The table values for the function are as follows:
x | y = x^(2/3)
--------------------
-10 | 4.6416
-5 | 2.924
-3 | 2.0801
-1 | 1
0 | 0
1 | 1
3 | 1.8171
5 | 2.924
10 | 4.6416
The domain of the function is all real numbers because any real number can be raised to the power of 2/3.
The range of the function is all non-negative real numbers, since the power of 2/3 ensures that the result will always be positive or zero.
Now, let's plot the points (-10, 4.6416), (-5, 2.924), (-3, 2.0801), (-1, 1), (0, 0), (1, 1), (3, 1.8171), (5, 2.924), and (10, 4.6416) on a graph and connect them to get a smooth curve.
The graph will be a concave upward curve starting from the origin (0, 0).
To draw the graph of the power function y = x^(2/3), we can plot some points and connect them to create a smooth curve. Here is a table of values for the function:
x | y
-----------
-8 | -4
-4 | -2
-2 | -1.587
-1 | -1
0 | 0
1 | 1
2 | 1.587
4 | 2
8 | 4
Now, let's plot these points on a coordinate plane:
[GRAPH WILL BE SHOWN HERE]
The domain of the power function y = x^(2/3) is all real numbers, since any real number can be raised to the 2/3 power.
The range of the power function is y ≥ 0, since the function outputs only nonnegative values.
To draw the graph of the power function y=x^(2/3), you can follow these steps:
1. Choose a set of x-values that cover a reasonable range. For example, you can choose x-values from -10 to 10.
2. Calculate the corresponding y-values by substituting each x-value into the function y=x^(2/3). For example, if x=-10, then y = (-10)^(2/3) ≈ 3.162. Repeat this process for each x-value you chose.
3. Plot the points (x, y) on a graph.
4. Connect the plotted points with a smooth curve. Since the exponent is less than 1, the curve will be steeper as x approaches 0.
Here is a table of values for the power function y=x^(2/3):
| x | y |
|-------|------------|
| -10 | 3.162 |
| -5 | 3.162 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 5 | 5 |
| 10 | 10 |
Regarding the domain and range of the function:
Domain: Since the function is defined for all real numbers, the domain is (-∞, ∞).
Range: The range of the function depends on the exponent. In this case, since the exponent is 2/3 (a fraction), the range will be all positive real numbers, including zero. Therefore, the range is [0, ∞).