Question

(a) Find functions from \(\mathbb{R}\) to \(\mathbb{R}\) that are: i. strictly decreasing ii. decreasing but not strictly decreasing iii. neither increasing nor decreasing iv. both increasing and decreasing (b) Show that no \(F: \mathbb{R} \rightarrow \mathbb{R}\) is both increasing and strictly decreasing. (c) Find a subset \(X \subseteq \mathbb{R}\) and a function \(F: X \rightarrow X\) where \(F\) is both strictly increasing and strictly decreasing.

Short answer

A constant function can be both increasing and decreasing but not strictly decreasing. No function can be both increasing and strictly decreasing. A function on a single-point set can be trivially both.

Step-by-step solution

Strictly Decreasing Function

A strictly decreasing function means that for any two points \(x_1 < x_2\), the function satisfies \(f(x_1) > f(x_2)\). One example of such a function is \(f(x) = -x\). This is because \(-x_1 > -x_2\) when \(x_1 < x_2\).

Decreasing But Not Strictly Decreasing Function

A function that is decreasing but not strictly decreasing allows for equal values in some cases: for any \(x_1 < x_2\), \(f(x_1) \geq f(x_2)\). A simple example is \(f(x) = -|x|\), as it remains constant at \(0\) for \(x=0\) and decreases otherwise.

Neither Increasing Nor Decreasing Function

A function is neither increasing nor decreasing if it does not consistently go up or down. A good example is \(f(x) = x^2\), because it decreases from negative infinity to zero then increases back to positive infinity.

Both Increasing and Decreasing Function

A constant function is both increasing and decreasing, as \(f(x_1) = f(x_2)\) for all \(x_1, x_2\). An example is \(f(x) = c\) for any constant \(c\).

Show that the Function Cannot Be Both Increasing and Strictly Decreasing

To be both increasing and strictly decreasing is logically impossible. If a function is strictly decreasing, it requires that \(f(x_1) > f(x_2)\) for \(x_1 < x_2\). Simultaneously, increasing requires \(f(x_1) \leq f(x_2)\), leading to a contradiction as it cannot satisfy both conditions simultaneously for any \(x_1 < x_2\).

Find a Subset X and Function F That is Both Strictly Increasing and Strictly Decreasing

If we restrict the domain to a single point, say \(X = \{a\}\), the function \(F:X \to X\), defined as \(F(x)=a\), satisfies both conditions trivially as there are no distinct points \(x_1\) and \(x_2\) to compare, rendering both concepts moot.

Key concepts

Strictly Decreasing Function

A strictly decreasing function is characterized by its steady decline as you move along the graph from left to right. This property means that for any pair of points along the function, if the first point has a smaller x-value than the second, the function value at the first point must be larger than at the second point. In simple terms, as the input values increase, the output values decrease continually.

For example, the function defined by the rule \( f(x) = -x \) is strictly decreasing. Here’s why: if you take two different numbers where \( x_1 < x_2 \), you'll find that \( -x_1 > -x_2 \). So, as you increase from \( x_1 \) to \( x_2 \), the corresponding function value decreases, illustrating the decreasing behavior.

• Strictly decreasing functions are always heading downhill and never plateau.
• There are no points where the function "stops" changing direction — it just keeps falling.

Increasing Function

An increasing function is where the function values either consistently go up or potentially stay the same without ever decreasing as the input value rises. However, unlike strictly increasing functions, they allow for "ties" in output values across different input values.

For any two points where \( x_1 < x_2 \), the function satisfies \( f(x_1) \leq f(x_2) \). It signifies that as you move from a lower x-value to a higher one, the function value never decreases.

• This function type maintains or ascends continuously.
• Situations like a straight horizontal line (constant function) also fall under this category.
• It is permissive, allowing some parts to "level out" as it climbs.

Decreasing Function

A decreasing function is one with a non-increasing trend in output values as you scan from left to right across its domain. Here, for any two input values where \( x_1 < x_2 \), the output satisfies \( f(x_1) \geq f(x_2) \). It means the function may drop or remain the same but never rise.

This category includes functions that might have portions of constant value but predominantly trend downward. A perfect example is the function \( f(x) = -|x| \), which not only decreases except at zero but also stays constant at \( f(0) = 0 \).

• Decreasing functions can have flat segments, making them not strictly decreasing.
• The output maintains or dips as it progresses.
• It's the perfect choice to model scenarios where values diminish or stay the same but never escalate.

Constant Function

A constant function is a special type of function that remains unchanged irrespective of the input value provided. In formal terms, a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is constant if for all values of x, the function has the same value, i.e., \( f(x) = c \) where \( c \) is a constant.

This means there is no variation or fluctuation across the entire range of x. The graph of such a function is a horizontal line parallel to the x-axis.

• A constant function is trivially both increasing and decreasing, as the output remains constant regardless of input changes.
• Perfect for describing scenarios where the output doesn’t depend on input changes.
• Mathematically, these functions are stable and predictable.

By definition, since there's no change, any two input values will always produce identical results, making it intriguing to be both increasing and decreasing, yet neither strictly so.