Question

What are the domain and range for `x^2 + 9y^2 = 25`?

Answer 1

`{("Domain", [-5,5] or {x||x|<=5}),("Range", [-5/3,5/3] or {y||y|<=5/3}):}`

Explanation:

The domain represents those `x` values for which the equation is defined; in other words, certain operations (such as division by zero or nth even roots of negative numbers) are not defined or are not defined within the real numbers. The range represents the `y` values which the equation can be evaluated to.

We can see this in practice by solving the above equation for `y`:

`x^2 + 9y^2 = 25`

`9y^2 = 25 - x^2`

`y^2 = (25-x^2)/9`

`y = +-sqrt(25-x^2)/3`

We start with the domain. Since there is a square root involved, we know that the square roots of negative values is not defined in the real number system. Thus, the radicand (the term inside of the square root) must be strictly non-negative:

`25 - x^2 >= 0`

`25 >= x^2`

`5 >= |x|`

The range comes from examining the possible values that the expression for `y` can take on. The greatest absolute value possible would come when the radicand is the largest possible value, or when `x = 0`. This would make the greatest value of `|y|` be `5/3`. Since `y` is defined in this relation as being both + and - versions of this value, this implies that `y` can range from `-5/3` to `5/3` in value.

Putting these together:

`{("Domain", [-5,5] or {x||x|<=5}),("Range", [-5/3,5/3] or {y||y|<=5/3}):}`

Side Note

You could visually see this by recognizing the original relation given is the equation for an ellipse with center at (0,0) and a semi-major axis of 5 and a semi-minor axis of `5/3`. By graphing it you can easily see the domain and range expressed by the extents of the graph:

graph{x^2+9y^2=25 [-6, 6, -2.5, 2.5]}

Answer 2

Domain: `+-a or +-5 or -5 <=x <=5 or [-5,5]`
Range: `+-b or +-5/3 or -5/3 <=y <=5/3 or [-5/3,5/3]`

Explanation:

`x^2+9y^2=25 or x^2/25 +(9y^2)/25=1` or

`x^2/5^2 +y^2/(5/3)^2=1` .This is an ellipse of equation

`x^2/a^2 + y^2/b^2 = 1` , here, `a= 5 , b=5/3`

The major axis `(2a)` is the longest diameter and

the minor axis `(2b)`is the shortest diameter (narrowest part).

Domain is `+-a or +-5 or -5 <=x <=5 or [-5,5]`

Range is `+-b or +-5/3 or -5/3 <=y <=5/3 or [-5/3,5/3]` [Ans]