Technicality of describing where sqrt(x) is increasing

In summary, when determining where the square root of x is increasing, the interval would be from 0 to infinity. However, there is some debate on whether to use (0, inf) or [0, inf) notation. While some may argue that the function is not "switching" from decreasing to increasing at zero, others may argue that the function is still monotonically increasing over the set [0, infinity). Ultimately, it depends on how you define monotonicity. In either case, it is important to note that x^2 is strictly increasing on the interval [0, infinity).
When describing where $sqrt{x}$ (square root of x) is increasing, it's from zero to infinity. But, do you say (0,inf) or [0, inf) ?

(I'm tutoring a student in pre-calc, and this came up. They don't know any calculus.)

In a situation like where is $x^2$ inc/dec, we'd say inc: (0, inf) and dec (-inf, 0). We wouldn't include the zero because that's the point where the function switches from dec to inc.

But on square root of x, it's not "switching" from dec to inc, so can we include that zero?

Thanks so much.

A function f is monotonically increasing over some set S if for any $x<y \Rightarrow f(x) < f(y) \, \forall x,y \in S$. No need for derivatives here! With this definition, you could say [tex]\sqrt x[/itex] is a monitonically increasing function over the set $S=[0,\infty)$.

D H said:
A function f is monotonically increasing over some set S if for any $x<y \Rightarrow f(x) < f(y) \, \forall x,y \in S$. No need for derivatives here! With this definition, you could say [tex]\sqrt x[/itex] is a monitonically increasing function over the set $S=[0,\infty)$.
And also, note that $x^2$ is strictly increasing on $[0, +\infty)$!

1. What is the definition of "increasing" for a mathematical function?

The term "increasing" refers to the behavior of a function as its input variable increases. In other words, as the input increases, does the output also increase or does it decrease?

2. How do you determine where sqrt(x) is increasing?

To determine where sqrt(x) is increasing, you can take the derivative of the function and set it equal to zero. Then, solve for x to find the critical points. These points will divide the x-axis into intervals. You can then test a point in each interval to see if the function is increasing or decreasing in that interval.

3. What is the mathematical notation for "sqrt(x) is increasing"?

The mathematical notation for "sqrt(x) is increasing" is f'(x) > 0, where f(x) is the function sqrt(x). This means that the derivative of the function is positive, indicating that the function is increasing.

4. Is it possible for sqrt(x) to be increasing and decreasing at the same time?

No, it is not possible for a function to be increasing and decreasing at the same time. A function can only have one behavior (increasing or decreasing) at any given point on its domain.

5. Can the technicality of describing where sqrt(x) is increasing be applied to other functions?

Yes, the technique of finding the critical points and testing points in each interval can be applied to any function to determine where it is increasing or decreasing. This technique is commonly used in calculus to analyze the behavior of functions.

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