what is the domain of h(x)=.002sec(3pi(x-.25))+27?
To determine the domain of the function h(x) = 0.002sec(3π(x - 0.25)) + 27, we need to consider any potential restrictions on the input values of x.
The domain of a function is the set of all allowable input values for which the function is defined. For h(x), we should consider any values of x that would result in an undefined secant function.
The secant function, sec(x), is undefined for values of x where cos(x) = 0. Since the secant function is the reciprocal of the cosine function, it produces undefined results when cosine is equal to zero.
The cosine function is equal to zero at π/2, 3π/2, 5π/2, and so on. Therefore, we need to find the values of x where 3π(x - 0.25) is equal to one of these values.
Let's set up an equation to find these values:
3π(x - 0.25) = π/2
Simplifying the equation:
3πx - 3π(0.25) = π/2
3πx - 3π/4 = π/2
3πx = π/2 + 3π/4
3πx = 2π/2 + 3π/4
3πx = 4π/4 + 3π/4
3πx = 7π/4
x = 7π/12
Similarly, we can find other values of x where the cosine is zero:
3π(x - 0.25) = 3π/2
x - 0.25 = 1/2
x = 1/2 + 0.25
x = 0.75
Therefore, the values x = 7π/12 and x = 0.75 are the critical points where the secant function becomes undefined.
Hence, the domain of the function h(x) = 0.002sec(3π(x - 0.25)) + 27 is all real numbers except x = 7π/12 and x = 0.75.