To find a piecewise formula for the given graph, we need to analyze the behavior of the function on different intervals.
Looking at the graph, we can see that the function consists of two semicircles. The semicircle on the left is centered at (-4, 0) with a radius of 4, and the semicircle on the right is centered at (4, 0) with a radius of 4.
Moreover, we are given that the domain of the function is [-8, 8]. This means that the function is defined for all values of x between -8 and 8, inclusive.
Let's break down the function into different cases based on the intervals:
1. For -8 <= x < -4:
In this interval, the function lies on the left semicircle. The equation of a circle centered at (h, k) with radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
For the left semicircle, we have:
(x + 4)^2 + y^2 = 4^2
Rearranging this equation, we get:
y = sqrt(16 - (x + 4)^2)
2. For -4 <= x <= 4:
In this interval, the y-values are always 0. So the function is simply y = 0.
3. For 4 < x <= 8:
In this interval, the function lies on the right semicircle. Using the same equation of a circle as before, the equation for the right semicircle is:
(x - 4)^2 + y^2 = 4^2
Rearranging this equation, we get:
y = sqrt(16 - (x - 4)^2)
Combining these cases, we can write the piecewise formula for the function f(x) as follows:
{ sqrt(16 - (x + 4)^2), -8 <= x < -4
f(x) = { 0, -4 <= x <= 4
{ sqrt(16 - (x - 4)^2), 4 < x <= 8
That's your piecewise formula for the given graph.