Suppose F ( X F(x F ( X ] Is A Function Such That If P \textless Q P \ \textless \ Q P \textless Q , Then F ( P ) \textless F ( Q F(p) \ \textless \ F(q F ( P ) \textless F ( Q ]. Which Statement Best Describes F ( X F(x F ( X ]?A. F ( X F(x F ( X ] Can Be Odd Or Even.B. F ( X F(x F ( X ] Can Be Odd But Cannot

Introduction

In mathematics, functions are used to describe the relationship between variables. A function is a rule that assigns to each input, or element, in a set, a unique output, or element, in another set. The properties of a function are crucial in understanding its behavior and characteristics. In this article, we will explore the properties of a function $f(x)$ that satisfies the condition: if $p < q$, then $f(p) < f(q)$. We will analyze the possible statements that describe the function $f(x)$ and determine which one is the best description.

The Given Condition

The given condition states that if $p < q$, then $f(p) < f(q)$. This means that as the input $x$ increases, the output $f(x)$ also increases. This is a fundamental property of a function, and it is known as the monotonicity of the function.

Monotonicity of a Function

A function $f(x)$ is said to be monotonic if it is either monotonically increasing or monotonically decreasing. A function is monotonically increasing if $f(p) < f(q)$ whenever $p < q$. A function is monotonically decreasing if $f(p) > f(q)$ whenever $p < q$.

Analyzing the Possible Statements

We are given two possible statements that describe the function $f(x)$:

A. $f(x)$ can be odd or even. B. $f(x)$ can be odd but cannot be even.

We need to analyze these statements and determine which one is the best description of the function $f(x)$.

Statement A: $f(x)$ can be odd or even

An odd function satisfies the condition $f(-x) = -f(x)$ for all $x$ in the domain of the function. An even function satisfies the condition $f(-x) = f(x)$ for all $x$ in the domain of the function.

If $f(x)$ is an odd function, then $f(-x) = -f(x)$. This means that $f(-x) < f(x)$ whenever $x > 0$, since $-x < x$ whenever $x > 0$. However, this contradicts the given condition that $f(p) < f(q)$ whenever $p < q$. Therefore, $f(x)$ cannot be an odd function.

If $f(x)$ is an even function, then $f(-x) = f(x)$. This means that $f(-x) = f(x)$ for all $x$ in the domain of the function. However, this does not necessarily mean that $f(x)$ is monotonically increasing or monotonically decreasing.

Statement B: $f(x)$ can be odd but cannot be even

We have already shown that $f(x)$ cannot be an odd function, since it contradicts the given condition. Therefore, this statement is not possible.

Conclusion

Based on our analysis, we can conclude that the best description of the function $f(x)$ is:

The function $f(x)$ is monotonically increasing.

This is because the given condition states that if $p < q$, then $f(p) < f(q)$. This means that as the input $x$ increases, the output $f(x)$ also increases. This is a fundamental property of a function, and it is known as the monotonicity of the function.

Properties of a Monotonically Increasing Function

A monotonically increasing function has several important properties:

• Monotonicity: The function is monotonically increasing, meaning that as the input $x$ increases, the output $f(x)$ also increases.
• No local maxima: A monotonically increasing function has no local maxima, since the function is always increasing.
• No local minima: A monotonically increasing function has no local minima, since the function is always increasing.
• Increasing derivative: The derivative of a monotonically increasing function is always positive, since the function is always increasing.

Examples of Monotonically Increasing Functions

Some examples of monotonically increasing functions include:

• Linear functions: A linear function of the form $f(x) = ax + b$ is monotonically increasing if $a > 0$.
• Quadratic functions: A quadratic function of the form $f(x) = ax^2 + bx + c$ is monotonically increasing if $a > 0$.
• Exponential functions: An exponential function of the form $f(x) = a^x$ is monotonically increasing if $a > 1$.

Conclusion

Introduction

In our previous article, we explored the properties of a function $f(x)$ that satisfies the condition: if $p < q$, then $f(p) < f(q)$. We concluded that the function $f(x)$ is monotonically increasing. In this article, we will answer some frequently asked questions about monotonically increasing functions.

Q: What is a monotonically increasing function?

A monotonically increasing function is a function that satisfies the condition: if $p < q$, then $f(p) < f(q)$. This means that as the input $x$ increases, the output $f(x)$ also increases.

Q: What are some examples of monotonically increasing functions?

Some examples of monotonically increasing functions include:

• Linear functions: A linear function of the form $f(x) = ax + b$ is monotonically increasing if $a > 0$.
• Quadratic functions: A quadratic function of the form $f(x) = ax^2 + bx + c$ is monotonically increasing if $a > 0$.
• Exponential functions: An exponential function of the form $f(x) = a^x$ is monotonically increasing if $a > 1$.

Q: What are the properties of a monotonically increasing function?

A monotonically increasing function has several important properties:

• Monotonicity: The function is monotonically increasing, meaning that as the input $x$ increases, the output $f(x)$ also increases.
• No local maxima: A monotonically increasing function has no local maxima, since the function is always increasing.
• No local minima: A monotonically increasing function has no local minima, since the function is always increasing.
• Increasing derivative: The derivative of a monotonically increasing function is always positive, since the function is always increasing.

Q: Can a monotonically increasing function have a local maximum?

No, a monotonically increasing function cannot have a local maximum. This is because the function is always increasing, and therefore, it cannot have a point where the function value is greater than the values at neighboring points.

Q: Can a monotonically increasing function have a local minimum?

No, a monotonically increasing function cannot have a local minimum. This is because the function is always increasing, and therefore, it cannot have a point where the function value is less than the values at neighboring points.

Q: What is the derivative of a monotonically increasing function?

The derivative of a monotonically increasing function is always positive. This is because the function is always increasing, and therefore, the derivative is always positive.

Q: Can a monotonically increasing function be differentiable?

Yes, a monotonically increasing function can be differentiable. In fact, the derivative of a monotonically increasing function is always positive.

Conclusion

In conclusion, a monotonically increasing function is a function that satisfies the condition: if $p < q$, then $f(p) < f(q)$. This means that as the input $x$ increases, the output $f(x)$ also increases. A monotonically increasing function has several important properties, including monotonicity, no local maxima, no local minima, and an increasing derivative.