Question

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Tosha is trying to find the solution(s) to the system {f(x)=x3−3x2+2g(x)=−x2+x.
After analyzing the graph of the functions, Tosha comes up with the following list of ordered pairs as possible solutions: (−1,−2), (0,0), (1,0), and (2,−2).

Part A: Which method should Tosha use to determine which of the ordered pairs are solutions to the system of equations?

Part B: What are the solutions to the system?

Select one answer for Part A, and select all answers that apply for Part B.

A: Tosha must verify each solution individually. For (−1,−2), she should evaluate f(−2) and g(−2). If she gets −1 as the result in both cases, then (−1,−2) is a solution of the system. She will test each of the other ordered pairs in the same way.
B: (0,0)
B: (2,−2)
A: Tosha must verify each solution individually. For (−1,−2), she should evaluate f(−1) and g(−2). If f(−1)=−2 and g(−2)=−1 then (−1,−2) is a solution of the system. She will test each of the other ordered pairs in the same way.
A: Tosha must verify each solution individually. For (−1,−2), she should evaluate f(−1) and g(−1). If she gets −2 as the result in both cases, then (−1,−2) is a solution of the system. She will test each of the other ordered pairs in the same way.
B: (−1,−2)
A: Tosha must verify each solution individually. For (−1,−2), she should evaluate f(−2) and g(−1). If f(−2)=−1 and g(−1)=−2 then (−1,−2) is a solution of the system. She will test each of the other ordered pairs in the same way.
B: (1,0)
A: Tosha can verify whether these ordered pairs are solutions by zooming in on the graph at each ordered pair to determine whether it is a solution.

I have tried multiple options and have got it incorrect I seem to be lost. If you can help it would be greatly apprieted

Answer 1

Not sure of your typing, I will assume you have:
f(x)=x^3−3x^2+2
g(x)=−x^2+x

So rather than all that wordage morass , why not just solve the system?
x^3−3x^2+2 = −x^2+x
x^3 + 2x^2 - x + 2 = 0

let's plot my interpretation:
https://www.wolframalpha.com/input/?i=plot+%E2%88%92x%5E3%2B3x%5E2+%3D%E2%88%92x%5E4%2B5x%5E3%E2%88%926x%5E2%2B3+

clearly, none of the answers found by Tosha will work, so you must mean something else.
You start with { but never close it.
Is f(x) = x^3−3x^2+2g(x) , where g(x) = −x^2+x

then you would simply have
f(x) = x^3−3x^2+2(−x^2+x)
and now you have a single function which looks like this:
https://www.wolframalpha.com/input/?i=plot+f%28x%29+%3D+x%5E3%E2%88%923x%5E2%2B2%28%E2%88%92x%5E2%2Bx%29+

Again, none of your given answers apply, so ....
check your question