Zeroes at 1±sqrt(2) ,y-intercept at –4

f(x)=?

many functions meet those criteria, but one of them is

given roots:
y = (x - (1+√2))(x-(1-√2))
= x^2-2x-1

That has a y-intercept of -1. So, multiply it by 4. It has the same roots, but now y(0) = -4.

To determine the equation of a function given its zeroes and the y-intercept, we need to use the concept of factoring.

First, let's start with the zeroes at 1±sqrt(2). This means that when x is equal to 1+sqrt(2) or 1-sqrt(2), the function evaluates to zero. To find the factors of the function, we subtract these values of x from both sides of the equation.

So, (x - (1+sqrt(2))) and (x - (1-sqrt(2))) are the factors of the function.

Next, let's find the y-intercept at -4. The y-intercept is the value of the function when x is equal to zero. Therefore, when x = 0, f(x) = -4.

Now, we can form the equation of the function by multiplying the factors we found and substituting the y-intercept:

f(x) = a * (x - (1+sqrt(2))) * (x - (1-sqrt(2)))

To find the value of 'a', we substitute the coordinates of the y-intercept:

-4 = a * (0 - (1+sqrt(2))) * (0 - (1-sqrt(2)))

Simplifying the equation:

-4 = a * (-1-sqrt(2)) * (-1+sqrt(2))

Expanding the equation further:

-4 = a * (1 - 2)

-4 = -a

Finally, solving for 'a':

a = 4

Therefore, the equation of the function f(x) is:

f(x) = 4 * (x - (1+sqrt(2))) * (x - (1-sqrt(2)))