SOLUTION: I am looking for the domain and the range for the equation: x^2+y = 10 I thought that the square root of a negative number was imaginary and, as such we could not have a functi

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Question 179180This question is from textbook College Mathematics
: I am looking for the domain and the range for the equation: x^2+y = 10
I thought that the square root of a negative number was imaginary and, as such we could not have a function from this equation. Apparently, this is not correct. Can you please explain what are and how to find the domain and range here? This question is from textbook College Mathematics

Found 3 solutions by Mathtut, jim_thompson5910, edjones:
Answer by Mathtut(3670)   (Show Source):
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write this as y=-x^2+10
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the domain (input values) x values in this case is all real numbers
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the range (output values) y values in this case is much more difficult to determine normally.
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try solving for x: this doesnt always work but at times it does, and is a good starting point.
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x= + or - since we cant have a negative number inside the square root sign ,this does limit the range. -y+10 has to be greater than or equal to zero.
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..........remember that when you divide or multiply an inequality by a negative number you need to reverse the sign.....
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so {y:y<=10} is the range...so all reals less than 10. y cannot for instance take on the value 12 otherwise you would have 12=-x^2+10
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2=-x^2
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x^2=-2...and we know that wont work...

Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Start with the given equation

Subtract from both sides.

Domain:

So we can see here that we can plug in any value for "x" and get a real number for "y". This means that the domain is all real numbers, which is in interval notation.

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Range:
Now there are different ways about going to find the range. You could graph the equation to find the set of y values. It turns out that this function graphs a parabola. Since parabolas have either a min or a max, this means that either the set of y values is either less than or equal to the max OR the set of y values is greater than or equal to the min.

In this case, since the coefficient for the term is -1, this means that the parabola will open down and that the parabola has a max. It turns out that the max value is 10. So the range is which in interval notation is (]

Answer by edjones(8007)   (Show Source):
You can put this solution on YOUR website!
y=ax^2+bx+c the form of a quadratic equation.
y=-x^2+10
a=-1, b=0 c=10
a is negative so the parabla has a maximum (opens downward).
The domain (of x) is all real numbers.
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The maximum is (-b^2-4ac)/4a
(-0^2-(4*-1*10))/4
=40/4=10
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The range (for y) is 10 to, but not including, -infinity or [10, -infinity)
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Ed
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