Find the Function f(x)=4x

$f (x) = 4 x$

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

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

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int x d x$

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

$4 (\frac{1}{2} x^{2} + C)$

Simplify the answer.

Rewrite $4 (\frac{1}{2} x^{2} + C)$ as $4 (\frac{1}{2}) x^{2} + C$ .

$4 (\frac{1}{2}) x^{2} + C$

Simplify.

Combine $4$ and $\frac{1}{2}$ .

$\frac{4}{2} x^{2} + C$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} x^{2} + C$

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} x^{2} + C$

Rewrite the expression.

$\frac{2}{1} x^{2} + C$

Divide $2$ by $1$ .

$2 x^{2} + C$

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

$F (x) = 2 x^{2} + C$