If Y=(X) has a range of [3,18], what is the range of Y =1/3f(x-2)-4?
I took the first 3-4=-1
Then the 18-4=14
R[-1,14]
But I got it wrong so can you please explain the steps
Well, let's break it down step by step:
1. The given function is Y = (X).
2. The range of Y = (X) is [3, 18]. This means that for any value of X, Y will be between 3 and 18.
3. Now, let's consider the new function Y = 1/3f(x-2)-4. We need to find the range of this function.
4. The first step is to substitute the range of Y = (X) into Y = 1/3f(x-2)-4. This will give us the new range.
5. Let's substitute the lower bound of the range, 3, into the equation: Y = 1/3f(x-2)-4. We get Y = 1/3f(3-2)-4, which simplifies to Y = 1/3f(1)-4.
6. Next, substitute the upper bound of the range, 18, into the equation: Y = 1/3f(x-2)-4. We get Y = 1/3f(18-2)-4, which simplifies to Y = 1/3f(16)-4.
7. Now, we have the new range for Y, which is determined by the values obtained using the lower and upper bounds of the original range. In this case, the new range is [1/3f(1)-4, 1/3f(16)-4].
It's important to note that we can't determine the exact values of 1/3f(1) and 1/3f(16) without more information about the function f. So, the range of Y can't be determined unless we have additional information about that particular function.
To find the range of the function Y = 1/3f(x-2) - 4 when the range of f(x) is [3,18], we need to apply a series of transformations and determine the new range.
1. Start with the given range of f(x): [3, 18].
2. First, consider the transformation f(x-2). Shifting the function f(x) two units to the right will also shift the range to the right by two units. So, the new range becomes [3+2, 18+2] which simplifies to [5, 20].
3. Next, we have the transformation 1/3f(x-2). Multiplying the entire range by 1/3 will scale down the range. Therefore, the new range becomes [(1/3) * 5, (1/3) * 20] which simplifies to [5/3, 20/3].
4. Finally, we subtract 4 from each value in the range. The new range becomes [5/3 - 4, 20/3 - 4], which simplifies to [-7/3, 8/3].
So, the range of Y = 1/3f(x-2) - 4 is [-7/3, 8/3].
To find the range of the function Y = (1/3)f(x-2) - 4, we need to consider the range of the original function Y = f(x).
Given that the range of Y = (X) is [3, 18], it means that the output values of the function (Y) lie between 3 and 18.
To determine the range of Y = (1/3)f(x-2) - 4, we need to apply each step of the function transformation to the original range.
1. Apply the shift: Since the input to the function is (x-2), the entire graph will shift 2 units to the right.
2. Apply the horizontal stretch: The function is multiplied by 1/3, which means it will be horizontally compressed. This compression will affect the range of the function.
3. Apply the vertical shift: The entire function is shifted downward by 4 units.
To find the range after each transformation:
**Step 1: Shift 2 units to the right**
The original range of [3, 18] will also shift 2 units to the right.
New range: [5, 20]
**Step 2: Horizontal compression by 1/3**
The range will be compressed horizontally by a factor of 1/3.
New range: [5/3, 20/3]
**Step 3: Shift downward by 4 units**
The entire range will be shifted downward by 4 units.
New range: [5/3 - 4, 20/3 - 4]
Simplifying,
New range: [-7/3, 8/3]
Therefore, the range of Y = (1/3)f(x-2) - 4 is [-7/3, 8/3].