SOLUTION: What is the domain and range of the function g(x)= x^4-3x^3+2x^2+x-1 I had gotten a domain of all real numbers, and I had estimated the range to include all numbers from [-1

Algebra ->  Functions -> SOLUTION: What is the domain and range of the function g(x)= x^4-3x^3+2x^2+x-1 I had gotten a domain of all real numbers, and I had estimated the range to include all numbers from [-1      Log On


   

Question 572760: What is the domain and range of the function g(x)= x^4-3x^3+2x^2+x-1
I had gotten a domain of all real numbers, and I had estimated the range to include all numbers from [-1, infinity), but I was told the range is incorrect.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
You're almost correct.
g%28x%29=x%5E4-3x%5E3%2B2x%5E2%2Bx-1=%28x-1%29%5E2%28x%5E2-x-1%29 has a double zero at x=1%7D%7D+and+two+other+zeros+at+%7B%7B%7Bx=%281+%2B-+sqrt%285%29%29%2F2, which are zeros of x%5E2-x-1.
It changes sign at x=%281+%2B-+sqrt%285%29%29%2F2, approximately -0.62 and 1.62, and is negative only in between those irrational zeros, but touches zero at x=1 in between.
g%280%29=-1 is close enough to the absolute minimum of the function, but so close to zero h%28x%29=2x%5E2%2Bx-1 is a good approximation, and since
h%28-0.1%29=2%2A0.01-0.1-1=-1.08 and h%28-0.2%29=2%2A0.04-0.2-1=-1.12, -1.1 may be a better estimate of the minimum.
With a calculator, or computer you could estimate it even closer.
I you were taught about derivatives of polynomial functions, you could exactly locate and calculate the minimum.