Question

Existence of an Inverse Determine the values of \(k\) such that the function \(f(x)=k x+\sin x\) has an inverse function.

Short answer

The function \(f(x) =kx+\sin(x)\) has an inverse for any non-zero real value of \(k\).

Step-by-step solution

Understand the Condition for Inverses

A function has an inverse if and only if it is strictly increasing or decreasing. This means that if \(k > 0\), then the function is increasing and if \(k < 0\), then the function is decreasing since the derivative of \(f(x)\) with respect to \(x\) is \(k + \cos(x)\).

Analyzing the function

\(\sin(x)\) is limited between 1 and -1. To ensure that \(f(x) =kx+\sin(x)\) is strictly increasing or decreasing, we should choose \(\cos(x)\) in combination with a non-zero \(k\) in a way that the sum is always positive for increasing (when \(k > 0\)) or negative for decreasing (when \(k < 0\)). Here, as \(\cos(x)\) ranges from -1 to 1, having \(k\) as a non-zero real number can always ensure that the derivative is greater than 0 (for \(k > 0\)) or less than 0 (for \(k < 0\)).

Determine the value of k

From the above analysis, to ensure the function has an inverse, \(k\) should be any non-zero real number.

Key concepts

Conditions for Inverses

Understanding when a function has an inverse is a fundamental concept in calculus. A prerequisite for a function to have an inverse is that it must be bijective; this means it must be both injective (one-to-one) and surjective (onto). To break it down further, for a function to have an inverse, it cannot map two different inputs to the same output, implying the function must be either strictly increasing or strictly decreasing.

Strict monotonicity ensures that each value of the output is paired with a unique input, thus avoiding any ambiguities when 'reversing' the function. Imagine trying to trace back a function that sometimes increases and sometimes decreases; finding the original input from an output could lead to multiple possibilities, which disrupts the one-to-one nature required for inverses. In the context of the exercise, determining values of 'k' such that the function f(x) = kx + sin(x) is strictly monotonic is key to identifying when the function possesses an inverse.

Strictly Increasing or Decreasing Functions

A strictly increasing function is one where, for any two inputs x1 and x2 with x1 < x2, the corresponding outputs satisfy f(x1) < f(x2). Similarly, a strictly decreasing function will have f(x1) > f(x2) for x1 < x2. These definitions are crucial as they provide a clear-cut rule for monotonicity that avoids the function ever 'turning back on itself'.

For the function in the exercise, f(x) = kx + sin(x), we look at the coefficient 'k' before the linear term 'x'. If 'k' is positive, adding sin(x) (which oscillates between -1 and 1) won’t prevent the overall function from increasing, since the slope of the linear term dominates the oscillation. If 'k' is negative, the same reasoning applies in the opposite direction, ensuring that the function decreases continuously. This concept ensures that with the right value of 'k', the function can be adjusted to maintain a consistent behavior, either always increasing or always decreasing, across its domain.

Derivative Analysis

The derivative of a function gives information about the function's rate of change at any given point. When analyzing a function for its potential to have an inverse, the derivative matters because it reveals whether the function is increasing or decreasing. A positive derivative across the function's entire domain signals that the function is increasing everywhere, while a negative derivative implies it is decreasing everywhere.

In our exercise, the derivative f'(x) = k + cos(x) must therefore be either always positive (if we want 'f' to be strictly increasing) or always negative (for 'f' to be strictly decreasing). Since cos(x) oscillates between -1 and 1, the magnitude of 'k' ensures that this oscillation does not change the sign of the derivative. If 'k' is selected correctly, f'(x) will not cross zero, ensuring the function preserves its strictly monotonic nature and thus can have an inverse.