The domain of a function, f(x), is all numbers between -1 and 6, inclusive. The range of this function is all numbers between 0 and 12, inclusive.
a) What is the domain of .5f(3x)?
b) What is the range of .5f(3x)?
domain: -1 <= 3x <= 6
so, -1/3 <= x <= 2
range = .5*range(f(x) = [0,6]
To determine the domain and range of .5f(3x), we need to consider the properties of the original function, f(x), and then apply the given transformations.
The domain of the function f(x) is all numbers between -1 and 6, inclusive. Therefore, any value of x within this interval will be valid for f(x).
Let's proceed with the calculations:
a) To determine the domain of .5f(3x), we need to consider the domain of f(x) and take into account the transformation of the function. In this case, the transformation is 3x, which scales the x-values of the function by a factor of 3 and shifts them accordingly.
Since the original domain of f(x) is from -1 to 6, we need to find the corresponding x values for the new function .5f(3x) by applying the transformation.
Applying the transformation, we get:
-1 * 3 = -3
6 * 3 = 18
Therefore, the new domain of .5f(3x) is all numbers between -3 and 18, inclusive.
b) To determine the range of .5f(3x), we need to consider the range of f(x) and apply the transformation.
The range of the function f(x) is all numbers between 0 and 12, inclusive.
Applying the transformation .5f(3x), the range will be affected by both the scaling factor of 0.5 and the original range of f(x).
Multiplying the original range by 0.5, we get:
0 * 0.5 = 0
12 * 0.5 = 6
Therefore, the new range of .5f(3x) is all numbers between 0 and 6, inclusive.
To find the domain and range of a function, we need to understand the given information and apply the appropriate transformations. Let's break it down step by step:
a) The domain of the function f(x), as given, is all numbers between -1 and 6, inclusive. To find the domain of .5f(3x), we need to consider the transformation that is applied to the function f(x). In this case, the function f(3x) represents a horizontal compression or stretching by a factor of 3.
To find the domain of .5f(3x), we need to divide the domain of f(x) by 3, as we want to scale it horizontally. Dividing the domain of -1 to 6 by 3 gives us the new domain of (-1/3) to (6/3), which simplifies to -1/3 to 2.
Therefore, the domain of .5f(3x) is all numbers between -1/3 and 2, inclusive.
b) The range of the function f(x), as given, is all numbers between 0 and 12, inclusive. To find the range of .5f(3x), we need to consider the transformation that is applied to the function f(x). In this case, the function .5f(3x) represents a vertical scaling by a factor of 0.5.
When you scale a range by a factor, it means multiplying the original range by that factor. Multiplying the range of 0 to 12 by 0.5 gives us the new range of 0 to 6.
Therefore, the range of .5f(3x) is all numbers between 0 and 6, inclusive.