Let y = f(x) be a function with domain D = [−6, −2] and range R = [−10, −4]. Find the domain D and range R for each function. Enter your answers using interval notation.
a) y = 1/2f(x)
b) y = f(x-2)+5
(a) clearly, the range is 1/2 the original range, and the domain is unchanged
(b) shift the domain down by 2, and the range up by 5
But, since f(x) is undefined for x outside the interval [-6,-2], the domain must be truncated.
To find the domain and range of the given functions, we need to understand how operations such as scaling, shifting, and adding constants affect the domain and range.
a) y = (1/2)f(x):
To find the domain of this function, we need to consider the domain of f(x) and any additional restrictions imposed by the operation. In this case, since f(x) has a domain D = [-6, -2], the function (1/2)f(x) will have the same domain.
Domain (a): D = [-6, -2]
To find the range of this function, we need to consider the range of f(x) and how the scaling operation affects it. The scaling factor of 1/2 does not affect the range directly; however, we need to consider if it affects the boundaries. As the function (1/2)f(x) scales down the values of f(x) by a factor of 1/2, the range will also be scaled down by the same factor.
Range (a): R = [(-10)*(1/2), (-4)*(1/2)] = [-5, -2]
b) y = f(x-2) + 5:
To find the domain of this function, we need to consider the domain of f(x-2). Shifting the input of f(x) by 2 units to the right will shift the domain of f(x) by 2 units to the right as well.
Domain (b): D = [-6+2, -2+2] = [-4, 0]
To find the range of this function, we need to consider the range of f(x) and how the shifting and adding operations affect it. Shifting the output of f(x) by 2 units to the right will not change the overall range as it is merely a horizontal shift. Adding 5 to the output will shift the range up by 5 units.
Range (b): R = [(-10+5), (-4+5)] = [-5, 1]
In summary:
a) Domain: D = [-6, -2]
Range: R = [-5, -2]
b) Domain: D = [-4, 0]
Range: R = [-5, 1]