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, ∞).