Express the volume of a sphere as a function of its radius. What is the domain of this function? Does this function have an inverse? If so what is it?
Volume of a sphere V=(4*Pi)/3*r^3
Domain all real numbers you can cube any real number and multiply it by another real number.
This is a one to one cubic with an inverse.
Solve for v in r=(4*Pi)/3*v^3 to find it.
It is v= (1/4)* (3^1/3)*(4^2/3)*(r/Pi)^1/3
The volume of a sphere can be expressed as a function of its radius (r) using the formula:
V(r) = (4/3)πr³
The domain of this function, in the context of a sphere, would be all positive real numbers for the radius, as a sphere can have a non-zero, positive radius.
Since the volume function is a one-to-one correspondence between the radius and the volume, it does indeed have an inverse. The inverse function can be found by solving the volume formula for the radius:
V(r) = (4/3)πr³
Divide both sides by (4/3)π:
(3/4π)V = r³
Take the cube root of both sides:
r = (3/4π)V^(1/3)
So, the inverse of the volume function is:
V^(-1)(V) = (3/4π)V^(1/3)
Where V^(-1) represents the inverse function of the volume function.
To express the volume of a sphere as a function of its radius, we can use the formula for the volume of a sphere:
V = (4/3)πr³
Where V represents the volume and r denotes the radius of the sphere.
Therefore, the volume function in terms of the radius is:
V(r) = (4/3)πr³
The domain of this function refers to the set of valid input values for the radius. In this case, since the radius represents a physical measurement, it must be a positive real number. Hence, the domain of the volume function is all positive real numbers (r > 0).
Now let's consider whether this function has an inverse. For a function to have an inverse, it must be a one-to-one function, meaning no two different inputs can have the same output.
In this case, since the volume function is V(r) = (4/3)πr³, notice that for every positive real number r, there is a unique volume value V. Therefore, the volume function is indeed one-to-one, and as a result, it has an inverse.
To find the inverse of the volume function, we can switch the roles of the input (radius) and output (volume) variables and solve for the radius.
Let y = (4/3)πx³
Swap x and y:
x = (4/3)πy³
Now, solve for y (which represents the radius):
y³ = (3/4π)x
y = (3/4π)^(1/3)x^(1/3)
Thus, the inverse of the volume function is:
V⁻¹(x) = (3/4π)^(1/3)x^(1/3)
where x represents the volume and y represents the radius.