Answer :
To solve this problem, we want to determine the more basic function that underwent transformations to become \( f(x)=-\frac{\sqrt{-x}}{3}+5 \).
Let's break down the process step-by-step:
Step 1: Start with the given function \( f(x) \).
[tex]\[ f(x) = -\frac{\sqrt{-x}}{3} + 5 \][/tex]
Step 2: Remove the transformations one by one to find the more basic form.
- Vertical translation: The function has a vertical shift of +5. By removing this shift, we have:
[tex]\[ f(x) - 5 = -\frac{\sqrt{-x}}{3} \][/tex]
- Vertical compression and reflection: The function includes a division by 3 and a negative sign, indicating a vertical compression by a factor of \( \frac{1}{3} \) and a reflection across the x-axis. To remove these, we multiply the entire function by -3:
[tex]\[ -3 \left(f(x) - 5\right) = \sqrt{-x} \][/tex]
- Square root and reflection within the argument: The function inside the square root has a negative sign, which indicates a reflection across the y-axis because \( -x \) inside the square root function reflects it horizontally.
Thus, the more basic function, after removing all transformations, can be expressed as:
[tex]\[ g(x) = -\sqrt{-x} \][/tex]
This is the basic function that, through a series of transformations, leads to the given function \( f(x) \).
So, the answer is:
[tex]\[ g(x)= -\sqrt{-x} \][/tex]
Let's break down the process step-by-step:
Step 1: Start with the given function \( f(x) \).
[tex]\[ f(x) = -\frac{\sqrt{-x}}{3} + 5 \][/tex]
Step 2: Remove the transformations one by one to find the more basic form.
- Vertical translation: The function has a vertical shift of +5. By removing this shift, we have:
[tex]\[ f(x) - 5 = -\frac{\sqrt{-x}}{3} \][/tex]
- Vertical compression and reflection: The function includes a division by 3 and a negative sign, indicating a vertical compression by a factor of \( \frac{1}{3} \) and a reflection across the x-axis. To remove these, we multiply the entire function by -3:
[tex]\[ -3 \left(f(x) - 5\right) = \sqrt{-x} \][/tex]
- Square root and reflection within the argument: The function inside the square root has a negative sign, which indicates a reflection across the y-axis because \( -x \) inside the square root function reflects it horizontally.
Thus, the more basic function, after removing all transformations, can be expressed as:
[tex]\[ g(x) = -\sqrt{-x} \][/tex]
This is the basic function that, through a series of transformations, leads to the given function \( f(x) \).
So, the answer is:
[tex]\[ g(x)= -\sqrt{-x} \][/tex]
