Question

1-6, show that \(f\) has an inverse by showing that it is strictly monotonic. $$ f(x)=-x^{5}-x^{3}-x $$

Short answer

The function is strictly decreasing and therefore has an inverse.

Step-by-step solution

Differentiate the Function

To determine if the function is monotonic, we first find its derivative. The function given is \( f(x) = -x^5 - x^3 - x \). The derivative of the function, \( f'(x) \), can be found using the power rule. Thus, \( f'(x) = -5x^4 - 3x^2 - 1 \).

Analyze the Sign of the Derivative

Now that we have the derivative \( f'(x) = -5x^4 - 3x^2 - 1 \), we examine its sign to determine if the function is strictly monotonic. Observe that \(-5x^4\), \(-3x^2\), and \(-1\) are all negative terms regardless of the value of \( x \). Thus, \( f'(x) \) is always negative for any real number \(x\).

Conclusion on Monotonicity

Since \( f'(x) < 0 \) for all \( x \), this indicates that the function is strictly decreasing. A strictly decreasing function is a strictly monotonic function.

Determine the Existence of the Inverse Function

A function that is strictly monotonic on its domain is both injective (one-to-one) and thus has an inverse function. Since we have shown that \( f(x) \) is strictly decreasing, we conclude that \( f(x) \) has an inverse.

Key concepts

Strictly Monotonic Functions

A function is considered strictly monotonic if it is either strictly increasing or strictly decreasing throughout its entire domain. This essentially means that the function consistently goes in one direction without changing its course. In other words, a strictly increasing function always goes upwards as you move to the right along the x-axis, while a strictly decreasing function continuously moves downwards.

For example, if a function is strictly monotonic, and specifically strictly decreasing like in the given exercise, it implies that each time the input grows, the output diminishes. This property is essential for proving that a function is injective because no two different inputs would yield the same output.

In our exercise, the function is shown to be strictly monotonic by noticing that its derivative has a consistent sign, indicating a unidirectional trend.

Derivative

A derivative provides the rate at which a function is changing at any given point and is particularly useful in determining monotonicity. In the context of our example, we are interested in finding the derivative of the function to discover how the function behaves across its entire range.

For the provided function, \( f(x) = -x^5 - x^3 - x \), using the power rule, we find that the derivative \( f'(x) = -5x^4 - 3x^2 - 1 \). Each term in the derivative \( -5x^4 \), \( -3x^2 \), and \( -1 \), includes a negative coefficient which signifies that as \( x \) changes, the derivative remains negative. This negative derivative confirms that the function is decreasing at each point, establishing that the function is strictly monotonic.

Understanding this concept is critical for analyzing function behaviors, especially in determining whether a function could be invertible based on monotonicity.

Monotonicity

Monotonicity refers to the consistent trend of a function within its domain: it could be either increasing, decreasing, or remaining constant at specific intervals. A function that consistently rises or falls is strictly monotonic, as opposed to one which stays the same over certain intervals or oscillates between increasing and decreasing.

In our example, the whole narrative focuses on demonstrating that the function is monotonic—specifically strictly decreasing. This is evident as we determined that the derivative \( f'(x) \) is always negative for all \( x \). Thus, the function’s monotonic trend is strictly downwards—which is essential for confirming the presence of an inverse.

Monotonicity ensures that a function remains injective, with no repeated values, critical for defining an inverse function.

Injective Function

Injective functions, often termed one-to-one functions, are those where each element in the domain maps to a unique element in the codomain. This means that no two different inputs map to the same output. Injectivity is crucial when determining if a function has an inverse.

For the function in our exercise, its derivation as strictly monotonic supports its injectivity. Because the strictly decreasing nature of the function denies any capacity for different inputs to yield the same output.

Given this injection, the function behaves in a manner that allows each input to result in a separate and unique output, which is what secures the formation of an inverse function. Therefore, the strictly decreasing trait of the function ensures that it’s injective, paving the path for its inverse to exist.