find the domain and range of y=f(x)
{4 if -6</= x< -2
f(x)={absolute value of x if -2</= x< 8
{cubed root of x if 8</= x</= 14
it is piecewise, I am completely lost on how to even set up this function.
the domain is all the values of x where f(x) is defined.
f(x) =
4 if -6 <= x < -2
|x| if -2 <= x < 8
∛x if 8 <= x <= 14
So, clearly the domain is [-8,14] since f(x) is defined everywhere inside the interval, and nowhere outside it.
Doesn't matter how piecewise it is. You just care about how it is defined. Just be sure to check that the pieces join up at the ends, leave no holes, and don't overlap.
To find the domain and range of the given piecewise function y=f(x), we need to determine the set of all possible values for x and the corresponding values for y.
1. Domain:
Looking at the given intervals, we can identify the range of x values for each segment of the function:
- Segment 1: -6 ≤ x < -2
- Segment 2: -2 ≤ x < 8
- Segment 3: 8 ≤ x ≤ 14
The domain of the function is the combination of all these intervals, which is:
Domain = [-6, -2) ∪ [-2, 8) ∪ [8, 14]
In interval notation, the domain is [-6, 14].
2. Range:
Now let's determine the range of y values for each segment of the function:
- Segment 1: f(x) = 4 for -6 ≤ x < -2
- Segment 2: f(x) = |x| for -2 ≤ x < 8
- Segment 3: f(x) = ∛x for 8 ≤ x ≤ 14
The range of the function is the combination of all these y values. Since there are no restrictions on the values that f(x) can take within each segment, the range is the set of all possible values for y.
Therefore, the range is:
Range = (-∞, +∞)
In interval notation, the range is (-∞, +∞).
To find the domain and range of the given piecewise function, we need to consider the validity of the function for different values of x.
First, let's consider the domain. The domain of a function is the set of all possible values of x for which the function is defined. In this case, the function is defined differently for different intervals of x.
For the first interval, -6 ≤ x < -2, the function f(x) equals 4. Therefore, the function is defined for all values of x within this interval.
For the second interval, -2 ≤ x < 8, the function f(x) equals the absolute value of x. This means that the function is defined for all values of x within this interval as well.
For the third interval, 8 ≤ x ≤ 14, the function f(x) is equal to the cubed root of x. Hence, the function is defined for all values of x within this interval.
Therefore, the domain of the function f(x) is given by the union of the domains in each piecewise function, which is:
Domain: -6 ≤ x ≤ 14
Now, let's move on to finding the range of the function. The range of a function is the set of all possible values taken by the function.
For the first interval, -6 ≤ x < -2, the function f(x) is always equal to 4. Therefore, the range consists of only one value, i.e., 4.
For the second interval, -2 ≤ x < 8, the function f(x) is equal to the absolute value of x. Absolute values are always non-negative, so the range for this interval is all non-negative values.
For the third interval, 8 ≤ x ≤ 14, the function f(x) is equal to the cubed root of x. The cubed root function is defined for all real numbers and can take on both positive and negative values.
Range: All non-negative numbers and all real numbers.
In summary, the domain of the function f(x) is -6 ≤ x ≤ 14, and the range includes all non-negative numbers and all real numbers.