let's call the line we're using to approximate f(x), g(x).

so g(x) = 2x-1.

then error = E = |f(x) - g(x)|

now f(x) - g(x) = x^2 - 2x + 3 - 2x + 1 = x^2 - 4x + 4 = (x - 2)^2

we want E ≤ 0.4, that is:

-0.4 ≤ (x - 2)^2 ≤ 0.4

since -0.4 < 0 < (x - 2)^2, for any x, we only need to check for:

(x - 2)^2 ≤ 0.4 = 4/10. taking square roots, we have:

x - 2 ≤ 2/√10

x ≤ 2 + 2/√10 = 2 + 2√10/10 = 2 + √10/5 = (10 + √10)/5

you might ask yourself...what is the smallest x could be? hint: it's NOT -(10 + √10)/5