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

$g (x) = 3 x - 2$

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.

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

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

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

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

Apply the constant rule.

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

Simplify.

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

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

Simplify.

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

Reorder terms.

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

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

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