Respuesta :
The domain of the function are all possible values for variable x and the range of the function are all possible values for the variable y.
Consider the parrent function [tex]y=\sqrt{x}.[/tex] The domain of this function is [tex]x\ge 0,[/tex] the range of this function is [tex]y\ge 0.[/tex]
1. Translate this function 5 units to the right. After this translation the function becomes
[tex]y=\sqrt{x-5}.[/tex]
2. Reflect previous function about the x-axis, then the expresssion of the new function is
[tex]y=-\sqrt{x-5}.[/tex]
The range of the function will be [tex]y\le 0.[/tex]
3. Translate the function [tex]y=-\sqrt{x-5}[/tex] 3 units up, then you will get the function
[tex]y=-\sqrt{x-5}+3.[/tex]
The domain of this function is [tex]x\ge 5,[/tex] the range of this function is [tex]y\le 3.[/tex]
Answer: [tex]y=-\sqrt{x-5}+3.[/tex]
You can use the fact that domain consists of those values for which function is well defined and range is the set of all values which the considered function outputs.
One of the function having the domain of x≥5 and a range of y≤3 is given by [tex]y = 3 - \sqrt{x-5}[/tex]
What is domain and range of a function?
- Domain is the set of values for which the given function is defined.
- Range is the set of all values which the given function can output.
How to find a function which has the domain of x≥5 and a range of y≤3 ?
If we do not consider complex numbers, we can take the square root as one of the operation which is undefined for the negative values. We can use that so that if input x goes below 5, that function becomes undefined.
Thus, we have [tex]\sqrt{x-5}[/tex] as a part such that any input below 5 will have undefined result. Since it is defined for all values bigger or equal to 5 for input, thus the domain is x≥5.
For the output to be only limited to 3 at max, we can use the fact that 3 - something non negative is always less or equal to 3. Since the expression [tex]\sqrt{x-5}[/tex] is non negative for any valid input, thus, we can use
[tex]y = 3 - \sqrt{x-5}[/tex]
Since at x = 5, the function outputs 3 and as x grows, the function's value decreases, and the function is not defined for x < 5 and doesn't output value > 3, thus, we have its domain of x≥5 and a range of y≤3
Thus,
One of the function having the domain of x≥5 and a range of y≤3 is given by [tex]y = 3 - \sqrt{x-5}[/tex]
Learn more about functions here:
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