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, with the points ordered such that their $x$-coordinates are in increasing order. (So "(1,-3); (2,3); (5,-7)" - without the quotes - is a valid answer format.)
To find the points where the graphs of the two functions intersect, we need to solve the equation $f(x) = g(x)$. In other words, we need to find the x-values that satisfy $x^3-x^2+x+1 = x^3+x^2+x-1$.
To solve this equation, we can subtract $x^3$ and subtract $x$ from both sides of the equation. This gives us $-x^2+2 = 0$.
Now we can solve for $x$. Adding $x^2$ to both sides, we get $x^2 = 2$. Taking the square root of both sides, we find $x = \pm \sqrt{2}$.
Therefore, the two points where the graphs of $f(x)$ and $g(x)$ intersect are approximately $(-\sqrt{2}, 2 \sqrt{2} - 1)$ and $(\sqrt{2}, -2 \sqrt{2} - 1)$.
So, the answer is $(-\sqrt{2}, 2 \sqrt{2} - 1); (\sqrt{2}, -2 \sqrt{2} - 1)$.
set the equations equal to each other, and solve for x (looks like two points)
plug the x values into either equation to find the y values of the intersection points