goo.gl/photos/e6Ks3v4Qqj177Nun6
The graph below is made from two semicircles. The domain of the function in the graph is [−8,8]
[-8,8]
. Find a piecewise formula for the function f(x)
f(x).
Help please, I already on my last attempt and any help is greatly appreciated!
a circle with center at (4,0) and radius 4 is
(x-4)^2 + y^2 = 16
or,
y = √(16-(x-4)^2)
The left semicircle has center at (-4,0) and radius 4, but lies below the x-axis, so
So, f(x) =
-√(16-(x+4)^2) for -4 <= x <= 0
√(16-(x-4)^2) for 0 <= x <= 4
Thanks Steve!
To find a piecewise formula for the function, we can analyze the graph provided.
At first glance, we can see that the graph is made up of two semicircles, one above the x-axis and one below.
We can divide the domain [-8, 8] into two intervals: [-8, 0] and [0, 8], corresponding to the left and right halves of the graph, respectively.
For the left half of the graph, we can see that it represents a semicircle centered at (-4, 0) with a radius of 4. The equation for the upper semicircle can be written as:
f(x) = √(16 - (x + 4)^2), -8 ≤ x ≤ 0
For the right half of the graph, we can observe that it represents a semicircle centered at (4, 0) with a radius of 4. The equation for the lower semicircle can be written as:
f(x) = -√(16 - (x - 4)^2), 0 ≤ x ≤ 8
Combining both equations, the piecewise formula for the function f(x) is:
f(x) = √(16 - (x + 4)^2), -8 ≤ x ≤ 0
-√(16 - (x - 4)^2), 0 ≤ x ≤ 8
Please note that the square root is the principal square root, denoted by √, ensuring that the function is non-negative.
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.