Find the Function g(x)=2x-3

$g (x) = 2 x - 3$

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

$G (x) = \int g (x) d x$

Split the single integral into multiple integrals.

$\int 2 x d x + \int - 3 d x$

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x d x + \int - 3 d x$

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

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

Apply the constant rule.

$2 (\frac{1}{2} x^{2} + C) - 3 x + C$

Simplify.

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Simplify.

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

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

$G (x) = x^{2} - 3 x + C$