Let f(x)=2.912345x^2+3.131579x-0.0
(a) to simplify approximation of f, write a quadratic function g(x) with integer coefficients that closely models f(x) for -10 < x < 10.

(b) compute g(4) and f (4)

(c) compute the error in using g (4) to approximate f(4) as a percentage of the correct answer f(4)

(d) for larger values of x (say x = 10 or x = 20), would g (x) be an overestimate or an underestimate of f (x)? justify your answer without computing specific values of f and g.

A.) Integers are positive and negative counting numbers. So, in order to find the integer coefficients, round off the coefficients in the equation to the nearest whole number. The function for g(x) is:

g(x) = 3x²+3x

B.) Substitute x=4 to the two functions.

f(x) = 2.912345x²+3.131579x-0.099999
f(4) = 2.912345(4)²+3.131579(4)-0.099999
f(4) = 59.023837

g(x) = 3x²+3x
g(4) = 3(4)²+3(4)
g(4) = 60

C.) The percentage error is equal to:

Percentage error = |g(4) - f(4)|/f(4)  * 100
Percentage error = |60 - 59.023837|/59.023837  * 100
Percentage error = 1.65%

D.) If x is a large number, for example x=10 or x=20, then g(x) would be an overestimate. This is because the value of x is raised to the power of 2. So, as the x increases, the corresponding function would increase exponentially. Even at x=4, g(x) is already an overestimate. What more for larger values of x? That means that the gap from the true answer f(x) would increase.

The answer would be 5< 6n