Find range of the function f(x) = sin (1/x)
range of any sine function is -1 to +1.
To find the range of the function f(x) = sin(1/x), let's consider the properties of the sine function and analyze the given function.
The sine function, sin(x), oscillates between -1 and 1. However, in our case, the input of the sine function is 1/x. To find the range, we need to determine the possible values of sin(1/x).
1. The input x is nonzero: When x is nonzero, 1/x will be defined, and sin(1/x) will take on all values between -1 and 1 as x approaches positive or negative infinity.
2. The input x approaches zero: As x approaches zero, 1/x becomes infinitely large in magnitude. When 1/x becomes large, sin(1/x) oscillates rapidly, taking on all values between -1 and 1. This means that sin(1/x) fills the entire range [-1, 1] as x approaches zero.
3. The input x equals zero: When x is exactly zero, 1/x is undefined, and sin(1/x) is not defined at this point. Hence, we exclude this value from the range.
Combining all these observations, we conclude that the range of the function f(x) = sin(1/x) is the interval [-1, 1], excluding the value 0.