A function f will be called repetitive...?

Question

A function f will be called repetitive if there are at least two different values of x in the interval 0 ≤ x ≤ 1 for which f(x) has the same value. What are all real numbers b for which f(x) = x^2 +bx + 3 is repetitive? Any help is appreciated, thanks.

Answer 1

I'll complete the square to make it clear:

f(x) = x² + bx + 3

f(x) = x² + bx + (b/2)² - (b/2)² + 3

f(x) = (x + b/2)² - b²/4 + 3

Okay if b = 0, we'll get the parabola f(x) = x² + 3 which has a vertex at (0, 0).  Notice that this parabola is not repetitive.  Now if we shift it over a bit (i.e. a shift which is less then 1 unit) you'll see that it becomes repetitive, but if you shift the parabola to the point that the vertex has an x value of 1, you'll see that once again that the parabola is no longer repetitive.  So what does this mean?  This means that the value we must horizontally translate it is greater then 0 but less then 1 unit.  Therefore, we have the following:

At the minimum:

b/2 = 0

b = 0

A the maximum:

b/2 = -1**

b = -2

Therefore, to guarantee that the function is repetitive we must have that -2 < b < 0.

**If you don't know why I have -1 it is because looking at the general expression:

f(x) = (x +  b/2)² - (b/2)² + 3

To create a horizontal shift to the right of 1 unit, b/2 MUST be negative otherwise it would be a shift to the left.

Hope this helps!