Question

What is the equation for a square root equation that increases over the interval [−2,∞)?

Answer 1

To find a square root function that increases over the interval \([-2, \infty)\), we can consider a general form of a square root function:

\[
f(x) = a \sqrt{x - h} + k
\]

Where:
- \(a\) is a positive constant to ensure the function increases,
- \(h\) is the horizontal shift,
- \(k\) is the vertical shift.

Since we want the function to be defined for the interval \([-2, \infty)\), we want the expression inside the square root, \(x - h\), to be non-negative starting from \(x = -2\).

Setting \(x - h \geq 0\) for \(x = -2\), we find:

\[
-2 - h \geq 0 \quad \Rightarrow \quad h \leq -2
\]

For simplicity, we can choose \(h = -2\). Thus, we have:

\[
f(x) = a \sqrt{x + 2} + k
\]

Here, the function will be defined for all \(x \geq -2\) and will increase if \(a > 0\).

So, for example, if we set \(a = 1\) and \(k = 0\), we get:

\[
f(x) = \sqrt{x + 2}
\]

This function \(f(x) = \sqrt{x + 2}\) is defined for \(x \geq -2\) and increases on the interval \([-2, \infty)\).

Therefore, one possible equation for a square root function that increases over the interval \([-2, \infty)\) is:

\[
f(x) = \sqrt{x + 2}
\]