Given the following function rules and domains, find the associated ranges.
(a) f(x) = 2x +3, Domain = {0, 1, 2}
(b) f(x) = 0, Domain = {0, 1, 2}
(c) f(x) = 2x + x2, Domain = {1, 0, 2}
Given a domain D, the range is the set {f(x) for all x element of D}
(a)
The range = {f(0), f(1), f(2)}.
Remember that sets need not contain duplicates.
To find the associated ranges for the given function rules and domains, we need to substitute the values in the domain into the function and evaluate the result. Let's solve each problem one by one:
(a) f(x) = 2x + 3, Domain = {0, 1, 2}
To find the range, we substitute each value in the domain into the function:
- For x = 0, f(0) = 2(0) + 3 = 0 + 3 = 3
- For x = 1, f(1) = 2(1) + 3 = 2 + 3 = 5
- For x = 2, f(2) = 2(2) + 3 = 4 + 3 = 7
Therefore, the associated range for this function is {3, 5, 7}.
(b) f(x) = 0, Domain = {0, 1, 2}
In this case, the function is a constant function, where f(x) always equals zero regardless of the value of x. Thus, the range for this function is {0}.
(c) f(x) = 2x + x^2, Domain = {1, 0, 2}
We substitute each value in the domain into the function:
- For x = 1, f(1) = 2(1) + 1^2 = 2 + 1 = 3
- For x = 0, f(0) = 2(0) + 0^2 = 0 + 0 = 0
- For x = 2, f(2) = 2(2) + 2^2 = 4 + 4 = 8
Therefore, the associated range for this function is {0, 3, 8}.