What are the coordinates of the points where f(x)=x^3-x^2+x+1
and g(x)=x^3+x^2+x-1 intersect?
To find the coordinates of the points where two functions intersect, we need to solve the equation f(x) = g(x). In this case, f(x) = x^3 - x^2 + x + 1 and g(x) = x^3 + x^2 + x - 1.
Setting f(x) equal to g(x), we have:
x^3 - x^2 + x + 1 = x^3 + x^2 + x - 1
Simplifying the equation, we get:
- x^2 + 2 = 0
To solve for x, we rearrange the equation:
x^2 = 2
Taking the square root of both sides, we obtain two possible solutions:
x = √2 or x = -√2
Substituting these values back into either f(x) or g(x) will give us the corresponding y-coordinates.
For example, substituting x = √2 into f(x) or g(x), we find:
f(√2) = (√2)^3 - (√2)^2 + √2 + 1
= 3 - 2 + √2 + 1
= √2 + 2
g(√2) = (√2)^3 + (√2)^2 + √2 - 1
= 3 + 2√2 - 1
= 2√2 + 2
Therefore, the coordinates of one point of intersection are (√2 + 2, 2√2 + 2).
Similarly, we can find the coordinates of the other point of intersection by substituting x = -√2.
Therefore, the coordinates of the points where f(x) and g(x) intersect are (√2 + 2, 2√2 + 2) and (-√2 + 2, -2√2 + 2).