Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function.
f(x) = cos [(3x)/2]
Find f'(x) and solve for f'(x)=0.
If f'(x) has one or more extrema, i.e. where f"(x)≠0, then the function is not monotonic, and an inverse is undefined on its entire domain.
For the case of f(x)=cos(x), there should be an infinite number of extrema.
To determine whether the function f(x) = cos [(3x)/2] is strictly monotonic on its entire domain, we need to find its derivative and check its sign.
First, find the derivative of f(x) using the chain rule.
f'(x) = -sin [(3x)/2] * (3/2)
Now, we need to check the sign of f'(x) to determine if f(x) is strictly monotonic.
The sign of f'(x) is determined by the sign of -sin [(3x)/2].
Since the range of the sine function is between -1 and 1, the sign of -sin [(3x)/2] will change as the angle (3x)/2 changes.
Now, we can observe that the factor (3/2) in the derivative does not affect the sign of -sin [(3x)/2], so we can ignore it for sign analysis.
Next, consider the sign of sin [(3x)/2]:
- When (3x)/2 = 0, sin [(3x)/2] = sin(0) = 0.
- When (3x)/2 = π/2, sin [(3x)/2] = sin(π/2) = 1.
- When (3x)/2 = π, sin [(3x)/2] = sin(π) = 0.
- When (3x)/2 = (3π)/2, sin [(3x)/2] = sin((3π)/2) = -1.
- When (3x)/2 = 2π, sin [(3x)/2] = sin(2π) = 0.
From the above observations, we can see that sin [(3x)/2] changes sign when (3x)/2 changes from 0 to π/2, π/2 to π, and π to (3π)/2. Therefore, sin [(3x)/2] is positive in the intervals (0, π/2) and (π, (3π)/2), and negative in the interval (π/2, π).
Since f'(x) = -sin [(3x)/2], this means that f'(x) is negative in the intervals (0, π/2) and (π, (3π)/2), and positive in the interval (π/2, π).
Based on the sign of f'(x), we can conclude that the original function f(x) = cos [(3x)/2] is strictly monotonic on its entire domain. Therefore, it has an inverse function.
To determine whether a function is strictly monotonic on its entire domain, we need to analyze its derivative. If the derivative of a function is positive (greater than zero) for all values in its domain, or negative (less than zero) for all values in its domain, then the function is strictly increasing or strictly decreasing, respectively, on its entire domain.
Let's find the derivative of the given function, f(x) = cos[(3x)/2]:
Step 1: Apply the chain rule.
The derivative of the cosine function is -sin.
The derivative of the inside function, (3x)/2, is 3/2.
Step 2: Multiply the derivatives we obtained in step 1 together.
The derivative of f(x) = cos[(3x)/2] is -sin[(3x)/2] * (3/2).
Now, we need to determine the sign of the derivative for the entire domain of f(x) = cos[(3x)/2].
To analyze the sign of the derivative, we can set it equal to zero and solve for x:
-sin[(3x)/2] * (3/2) = 0
From this equation, we can see that the derivative is equal to zero when sin[(3x)/2] = 0.
It means that the function has stationary points (or points of non-monotonic behavior) where sin[(3x)/2] = 0.
To solve sin[(3x)/2] = 0, we set the inside function equal to integer multiples of pi:
(3x)/2 = k*pi, where k is an integer.
Solving for x, we get:
x = (2k*pi)/3, where k is an integer.
So, the function has stationary points at x = (2k*pi)/3, where k is an integer.
Now, let's analyze the sign of the derivative in different intervals of the domain to determine monotonicity:
1. For x < (2k*pi)/3:
- sin[(3x)/2] is negative because it corresponds to negative y-values of the cosine function.
- The derivative, -sin[(3x)/2] * (3/2), is positive because we are multiplying a negative number by a positive number.
2. For x > (2k*pi)/3:
- sin[(3x)/2] is positive because it corresponds to positive y-values of the cosine function.
- The derivative, -sin[(3x)/2] * (3/2), is negative because we are multiplying a positive number by a negative number.
Based on our analysis, we can conclude that the function f(x) = cos[(3x)/2] is not strictly monotonic on its entire domain. It has alternating regions of increasing and decreasing behavior due to the stationary points. Therefore, it does not have an inverse function on its entire domain.