The function f has a domain of [0,5]and a range of [0,3]. Start by sketching a potential graph of f.
Suppose the function k is defined as k(x)=f(x−3). Determine the domain and range of k.
Domain:
Range:
the graph is shifted to the right by 3, so
domain is [3,8]
range is the same
Thank you
Domain: The domain of k would be the set of all values that can be plugged into k(x) without resulting in undefined or imaginary numbers.
In this case, since f has a domain of [0,5], any value of x can be plugged into f(x-3) as long as x-3 falls within the domain of f, which is [0,5]. Therefore, the domain of k would be [3,8].
Range: The range of k represents the set of all possible output values of k(x). Since we know that f has a range of [0,3], we can determine the range of k by considering the possible values of f(x-3) within that range.
When we subtract 3 from x, it shifts the domain of f three units to the right. This means that the range of k would be [0,3], the same as the range of f.
So, the domain of k is [3,8] and the range of k is [0,3].
To sketch a potential graph of f with a domain of [0,5] and a range of [0,3], we can start by labeling the x-axis from 0 to 5 and the y-axis from 0 to 3. Given that the domain of f is [0,5], we know that the graph of f will only exist within that x-range.
Since the range of f is [0,3], the graph will only pass through y-values between 0 and 3.
Now, let's determine the domain and range of k(x) = f(x - 3), where f is our original function.
Domain of k(x):
To find the domain of k(x), we need to consider the x-values that we can plug into k(x) without encountering any division by zero or undefined operations.
In this case, when we subtract 3 from x in f(x - 3), it means that we shift the graph of f three units to the right. Therefore, the domain of k(x) will be the values of x that would have been in the original domain [0,5] plus 3.
So, the domain of k(x) will be [0 + 3, 5 + 3], which simplifies to [3, 8].
Domain of k(x): [3, 8]
Range of k(x):
The range of k(x) will be determined by the range of f(x - 3). Since we have determined that the range of f is [0,3], when we shift the graph three units to the right, the range of k(x) will remain the same.
Therefore, the range of k(x) will also be [0,3].
Range of k(x): [0, 3]
To sketch a potential graph of function f, we know that the domain is [0,5] and the range is [0,3]. This means that the input for the function f can be any value between 0 and 5, inclusive, and the output can be any value between 0 and 3, inclusive.
To start the sketch, draw a set of x and y axes. Label the x-axis from 0 to 5 and the y-axis from 0 to 3. Since the range of f is [0,3], draw a horizontal line segment on the graph, starting from the origin (0,0) and extending to the right until x=5, and then going straight up until the y-coordinate reaches 3. The graph of f should be a rectangular region in the first quadrant with a height of 3 and a width of 5.
Now let's move on to determining the domain and range of function k, which is defined as k(x) = f(x-3).
To find the domain of k, we need to consider the values that can be input into the function k(x). In this case, the input for k(x) is (x-3). Since f has a domain of [0,5], we need to determine the values of x-3 that fall within this domain.
Setting the lower bound of f's domain equal to x-3, we have:
0 = x-3
x = 3
Setting the upper bound of f's domain equal to x-3, we have:
5 = x-3
x = 8
Therefore, the domain of k is [3,8].
To determine the range of k, we need to consider the output values of k(x) given the domain [3,8]. From the graph of f, we can see that the output values of f(x) were between 0 and 3. Therefore, the range of k will be the same range [0,3].
Thus, the domain of k is [3,8] and the range of k is [0,3].