To find the y value when f(x) = g(x), we need to set the two equations equal to each other and solve for x. Then, we can substitute the value of x into either f(x) or g(x) to find the corresponding y value.
Setting f(x) = g(x):
2x^3 + 2x - 3 = -0.5|x - 4|
To simplify the equation, we can consider two cases based on the absolute value:
Case 1: x - 4 is positive or zero (x - 4 ≥ 0)
In this case, we can remove the absolute value and rewrite the equation as:
2x^3 + 2x - 3 = -0.5(x - 4)
Case 2: x - 4 is negative (x - 4 < 0)
In this case, we need to flip the sign of the absolute value and rewrite the equation as:
2x^3 + 2x - 3 = -0.5(-x + 4)
Now, we can solve each case separately to find the possible values of x.
Case 1:
2x^3 + 2x - 3 = -0.5(x - 4)
2x^3 + 2x - 3 = -0.5x + 2
Combine like terms:
2x^3 + 2.5x - 5 = 0
We can use numerical or graphical methods to find the solutions for this cubic equation. Once we have the values of x, we can substitute them into either f(x) or g(x) to find the corresponding y values.
Case 2:
2x^3 + 2x - 3 = -0.5(-x + 4)
2x^3 + 2x - 3 = 0.5x - 2
Combine like terms:
2x^3 - 1.5x + 1 = 0
Again, we can use numerical or graphical methods to find the solutions for this cubic equation. Once we have the values of x, we can substitute them into either f(x) or g(x) to find the corresponding y values.
It's important to note that the equations f(x) = 2x^3 + 2x - 3 and g(x) = -0.5|x - 4| may not intersect at the same x values. Therefore, the y values when f(x) = g(x) will depend on the solutions we find for the equations above.