Which radical equations produce imaginary values of y for the given values of x?
I don't understand this can someone please explain?
Choices
y=√x, when x<0
y=√4−x, when x≤4
y=√4−x, when x>4
y=√x+8, when x≥−8
√x+8 , when x<−8
y=√x, when x≥0
y=√3x−6, when x<2
y=√3x−6, when x≥2
The answers are: options a, c, e, and g. I already took a test and got 100% with these answers.
remember that whatever is under the radical √ cannot be negative
so,
y=√x, when x<0 is not real
y=√4−x, when x>4 is not real, since if x>4, 4-x < 0
and so on for the others
Thank you :)
To understand which radical equations produce imaginary values of y for given values of x, we need to look at the domain (possible values) of x for each equation.
An imaginary number is a number that can be written as a real number multiplied by the imaginary unit "i," where i is the square root of -1. Imaginary numbers are denoted by the letter "i" in equations.
Let's evaluate each equation and determine the values of x that would result in imaginary values of y.
1) y=√x, when x<0:
In this equation, the square root of a negative number is always imaginary. So, all values of x less than zero (x<0) will produce imaginary values of y.
2) y=√4−x, when x≤4:
Here, if x is less than or equal to 4, we need to consider the expression inside the square root. When this expression becomes negative, the result will be imaginary. In this case, x≤4 includes the values x=4 and x less than 4, which will produce imaginary values of y.
3) y=√4−x, when x>4:
The inequality x>4 means x is strictly greater than 4. Since the expression inside the square root is 4-x, when x is greater than 4, this expression becomes negative. Therefore, for x>4, we will have imaginary values of y.
4) y=√x+8, when x≥-8:
In this equation, we have x values greater than or equal to -8. For x≥-8, the expression inside the square root (x+8) will represent non-negative values, which means there will be no imaginary values of y.
5) √x+8, when x<-8:
Here, we have x values less than -8. When x<-8, the expression inside the square root (x+8) becomes negative, resulting in imaginary values for y.
6) y=√x, when x≥0:
In this equation, x values greater than or equal to 0 will produce non-negative (real) values of y. There will be no imaginary values.
7) y=√3x−6, when x<2:
For x<2, the expression inside the square root (3x-6) will be negative, leading to imaginary values of y.
8) y=√3x−6, when x≥2:
The inequality x≥2 includes values equal to or greater than 2. For x≥2, the expression inside the square root (3x-6) will represent non-negative values, so there will be no imaginary values.
So, the radical equations that produce imaginary values of y for given values of x are:
- y=√x, when x<0
- y=√4−x, when x≤4
- y=√3x−6, when x<2
- √x+8, when x<-8