Find the Function f(x) = square root of x-3

$f (x) = \sqrt{x} - 3$

The function $F (x)$ can be found by evaluating the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Split the single integral into multiple integrals.

$\int \sqrt{x} d x + \int - 3 d x$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int x^{\frac{1}{2}} d x + \int - 3 d x$

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$\frac{2}{3} x^{\frac{3}{2}} + C + \int - 3 d x$

Apply the constant rule.

$\frac{2}{3} x^{\frac{3}{2}} + C - 3 x + C$

Simplify.

$\frac{2}{3} x^{\frac{3}{2}} - 3 x + C$

The function $F$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$F (x) = \frac{2}{3} x^{\frac{3}{2}} - 3 x + C$