What are the coordinates of the points where the graphs of f(x)=x^3-x^2+x+1 and g(x)=x^3+x^2+x-1 intersect? Give your answer as a list of points separated by semicolons. (So "(1,-3); (2,3); (5,-7)" is a valid answer format.)
To find the points where two graphs intersect, we need to set the equations equal to each other and solve for x. In this case, we have:
x^3 - x^2 + x + 1 = x^3 + x^2 + x - 1
By subtracting x^3 from both sides and canceling out the x^2 terms, we get:
- x^2 + 2 = 0
Adding x^2 to both sides, we have:
x^2 = 2
Taking the square root of both sides, we get:
x = ±√2
So the x-coordinates of the intersecting points are √2 and -√2.
To find the corresponding y-coordinates, we substitute these x-values into either equation. Let's use f(x) = x^3 - x^2 + x + 1:
For x = √2:
f(√2) = (√2)^3 - (√2)^2 + √2 + 1 = 2√2 - 2 + √2 + 1 = 3√2 - 1 ≈ 3.24
For x = -√2:
f(-√2) = (-√2)^3 - (-√2)^2 - √2 + 1 = -2√2 - 2 - √2 + 1 = -3√2 - 1 ≈ -4.24
Therefore, the points where the graphs of f(x) and g(x) intersect are (√2, 3.24) and (-√2, -4.24).
well, they intersect where f(x) = g(x).
x^3-x^2+x+1 = x^3+x^2+x+1
-x^2 = x^2
x = 0
to check, see
https://www.wolframalpha.com/input/?i=x^3-x^2%2Bx%2B1+%3D+x^3%2Bx^2%2Bx%2B1