Did you know?
Did you know that the domain of a function represents all the possible values of x that the function can take? In the given function f(x), the domain is {x∈R|−4 ≤ x ≤ 16}, which means x can be any real number between -4 and 16.
Similarly, the range of a function represents all the possible values of y that the function can output. In the given function f(x), the range is {y∈R|−8 ≤ y ≤ 12}, indicating that y can be any real number between -8 and 12.
Now, let's consider the function g(x) = 34f(−x+3) + 5. To find the domain and range of g(x), we need to analyze the effect of each operation on the original domain and range.
The function g(x) first involves the operation of -x+3, which reflects the original function across the y-axis and then shifts it to the right by 3 units. This operation does not affect the domain, so the domain of g(x) remains the same as the domain of f(x): {x∈R|−4 ≤ x ≤ 16}.
Next, g(x) multiplies the resulting function by 34 and adds 5. These operations do not affect the domain, so the domain of g(x) remains {x∈R|−4 ≤ x ≤ 16}.
Therefore, the domain of g(x) is {x∈R|−4 ≤ x ≤ 16}, which means x can be any real number between -4 and 16.
Similarly, these operations do not affect the range. Therefore, the range of g(x) is {y∈R| −8 ≤ y ≤ 12}, indicating that y can be any real number between -8 and 12.
