Question

Find two functions defined implicitly by this equation.
(X-4)^2 + (y + 3)^2 = 1

Answer 1

The equation given is that of a circle and solving for y in terms of x, we find two functions representing the upper and lower halves of the circle: y₁(x) = -3 + √(1 - (x - 4)²) and y₂(x) = -3 - √(1 - (x - 4)²).

Step-by-step explanation:

To find two functions defined implicitly by the equation (X-4)² + (y + 3)² = 1, we recognize that this equation represents a circle with a radius of 1, centered at (4, -3). We can solve for y as a function of x in two ways, corresponding to the upper and lower halves of the circle.

First, express y explicitly:

• y = -3 ± √(1 - (x - 4)²)

For the upper half:

• y₁(x) = -3 + √(1 - (x - 4)²)

For the lower half:

• y₂(x) = -3 - √(1 - (x - 4)²)

These two functions, y₁(x) for the upper part and y₂(x) for the lower part, are the solutions in terms of x.