okay i need to find the domain and range of this

z(x) = �ãx^2 - 100

(everything is in the square root)

so i think i found the domain i got [-10,10]

but i don't know about the range

i solved for x and got stuck here

x=�ãy^2 + 100

helppp pleaseee
:)

those wierd things were supposed to be square roots btw

To find the domain and range of the function z(x) = √(x^2 - 100), let's start with the domain.

Domain:
The square root function (√) requires the expression inside the square root to be non-negative. Therefore, x^2 - 100 must be greater than or equal to zero.

By solving the inequality x^2 - 100 ≥ 0, we can find the values of x that make the expression non-negative:

x^2 - 100 ≥ 0
(x - 10)(x + 10) ≥ 0

From this inequality, we can see that the expression is equal to zero at x = -10 and x = 10, and it changes sign at those points. Hence, the intervals where x^2 - 100 is non-negative are (-∞, -10] and [10, ∞). This means the domain of z(x) is:

Domain: (-∞, -10] ∪ [10, ∞)

Now let's move on to finding the range.

To find the range, we need to solve the equation x = √(y^2 + 100) for y.

Square both sides to eliminate the square root:

x^2 = y^2 + 100

Rearrange the equation to solve for y:

y^2 = x^2 - 100

Taking the square root of both sides gives:

y = ±√(x^2 - 100)

Since we are working with the positive square root, we can write this as:

y = √(x^2 - 100)

From this, we can see that the range of z(x) is the set of all non-negative real numbers, since the square root of any non-negative number will always yield a non-negative result.

Range: [0, ∞)

Therefore, the domain of z(x) is (-∞, -10] ∪ [10, ∞) and the range is [0, ∞).