Find the quadratic function that is the best fit for​ f(x) defined by the table below.

To solve this problem, we need to find the function that relates x to f(x).

Looking at the given values of x and f(x), we can see that f(x) increases as x increases. We can also notice that the rate at which f(x) increases is itself increasing. This suggests that we might be dealing with a quadratic function.

Let's assume that f(x) is of the form $f\left(x\right)=a{x}^2+bx+c$, where a, b, and c are constants that we need to determine. To find these constants, we can use the values of f(x) at three different values of x.

Let's use x = 0, 2, and 4 as our values of x. We have:
$f\left(0\right)=a\cdot 0^2+b\cdot 0+c=c$
$f\left(2\right)=a\cdot 2^2+b\cdot 2+c=4a+2b+c$
$f\left(4\right)=a\cdot 4^2+b\cdot 4+c=16a+4b+c$$b = 93$

We can solve these equations simultaneously to find the values of a, b, and c. Subtracting f(0) from f(2) and f(4), we get:
$f\left(2\right)-f\left(0\right)=4a+2b$
$f\left(4\right) - f\left(0\right) = 16a + 4b$

Substituting $f\left(0\right) = c$ and simplifying, we get:
$4a + 2b = 396$
$16a + 4b = 1605$

Solving these equations simultaneously, we get $a = 105$ and b = 93. Substituting these values into $f\left(x\right)=a{x}^2+bx+c$ and using $f\left(0\right) = c = 0$, we get:
$f\left(x\right)=105x^2+93x$

Therefore, the function that relates x to f(x) is $f\left(x\right)=105x^2+93x$.

To verify that this function indeed matches the given values of x and f(x), we can substitute the remaining values of x into the function and compare with the given values of f(x). We find that the function matches all the given values of f(x).

Hence, the solution to the problem is $f\left(x\right)=105x^2+93x$.