Question

Describe the interval(s) on which the function is continuous. $$ f(x)=\frac{x+1}{\sqrt{x}} $$

Short answer

The function is continuous on the interval (0, ∞). Notice that the end point 0 is not included in the interval since the function is not defined at x = 0.

Step-by-step solution

Identify the Domain

Since the denominator of the function includes a square root, the function is only defined for values where x>0. The square root of any number less than or equal to zero will result in an imaginary number, so the function is not defined for x≥0. Therefore, the function's domain is \(x > 0\)

Check for Continuity

Within its domain (x>0), the function is a ratio of two polynomial functions. Polynomial functions are known to be continuous on their domain, therefore, the ratio will be continuous on its domain, unless the denominator is zero. Since the square root of x is never zero for x>0, the function \(f(x) = \frac{x+1}{\sqrt{x}}\) doesn't have points of discontinuity x within its domain.

Formulate the Answer

After finding the function's domain and checking for points of discontinuity, it can be concluded that the function is continuous for all x in its domain.