The y-values when f(x) = g(x) are y = 1 for x = 1 and y = -5 for x = -1.
To find the y-value when f(x) = g(x), we need to solve the equation:
2x³ + 2x - 3 = -0.5|x-4|
First, we need to consider the absolute value function. Since the absolute value of an expression is its distance from zero, it has two solutions for any non-zero value: one positive and one negative. Therefore, we need to split the equation into two cases and solve each one:
Case 1: x - 4 ≥ 0
In this case, the absolute value function simplifies to x - 4. So, the equation becomes:
2x³ + 2x - 3 = x - 4
Combining like terms:
2x³ + x - 1 = 0
This can be factored as:
(x - 1)(2x² + x + 1) = 0
Therefore, x = 1 or x = (-1 ± √3i) / 2
Case 2: x - 4 < 0
In this case, the absolute value function simplifies to -(x - 4). So, the equation becomes:
2x³ + 2x - 3 = -(-x + 4)
Combining like terms:
2x³ + 3x + 1 = 0
This can be factored as:
(x + 1)(2x² - 2x + 1) = 0
Therefore, x = -1 or x = (1 ± √3i) / 2
Now, we need to check which solutions are valid for the original equation. We only want the solutions where f(x) = g(x).
When x = 1, both f(x) and g(x) are equal to 1. So, this is a valid solution.
When x = (-1 ± √3i) / 2, f(x) becomes a complex number, while g(x) remains real. Therefore, these solutions are not valid.
When x = -1, both f(x) and g(x) are equal to -5. So, this is also a valid solution.
Therefore, the y-values when f(x) = g(x) are:
1. y = 1 (when x = 1)
2. y = -5 (when x = -1)
So, the answer is:
y = 1 or y = -5