Problem 100
Question
Proof Use the definitions of increasing and decreasing functions to prove that $$ f(x)=\frac{1}{x} $$ is decreasing on \((0, \infty)\)
Step-by-Step Solution
Verified Answer
The function \(f(x)=\frac{1}{x}\) is decreasing on \((0, \infty)\) because for any two numbers \(x_{1}\) and \(x_{2}\) in that interval with \(x_{1} < x_{2}\), \(f(x_{1}) > f(x_{2})\).
1Step 1: Define the function and the interval
The function \(f(x)=\frac{1}{x}\) is evaluated over the interval \((0, \infty)\). We want to show that for any \(x_1, x_2\) in \((0, \infty)\) with \(x_1 < x_2\), \(f(x_1) > f(x_2)\).
2Step 2: Apply the function to numbers in the interval
For any \(x_1, x_2\) in the interval \((0, \infty)\) with \(x_1 < x_2\), apply the function to get \(f(x_1) = \frac{1}{x_1}\) and \(f(x_2) = \frac{1}{x_2}\).
3Step 3: Manipulate the inequality
We know that if \(x_1 < x_2\) then, because we are dealing with positive numbers, \(\frac{1}{x_1} > \frac{1}{x_2}\). From this, we conclude that \(f(x_1) = \frac{1}{x_1} > \frac{1}{x_2} = f(x_2)\).
4Step 4: Conclude the proof
Since \(f(x_1) > f(x_2)\) for all \(x_1, x_2\) with \(x_1 < x_2\), we can say that the function \(f(x) = \frac{1}{x}\) is decreasing on the interval \((0, \infty)\).
Key Concepts
Increasing FunctionsProof by DefinitionFunction BehaviorInterval Analysis
Increasing Functions
In mathematics, a function is called an increasing function if as one variable increases, the function value also increases.The formal definition of an increasing function states that a function \( f(x) \) is increasing on an interval if for any two numbers \( x_1 \) and \( x_2 \) within that interval, whenever \( x_1 < x_2 \), it follows that \( f(x_1) \leq f(x_2) \).
This means the graph of an increasing function rises as it moves from left to right.
Examples of increasing functions include functions like \( f(x) = x^2 \) for \( x \geq 0 \), where the function value becomes larger as \( x \) increases. It's essential to determine the interval over which you're evaluating to ensure the function maintains its increasing nature within that specific context.
This means the graph of an increasing function rises as it moves from left to right.
Examples of increasing functions include functions like \( f(x) = x^2 \) for \( x \geq 0 \), where the function value becomes larger as \( x \) increases. It's essential to determine the interval over which you're evaluating to ensure the function maintains its increasing nature within that specific context.
Proof by Definition
Proving properties of a function, such as whether it's increasing or decreasing, often requires using the definition of these properties directly.In the case of decreasing functions, we use the definition that a function \( f(x) \) is decreasing on an interval if, for every pair \( x_1 \) and \( x_2 \) in the interval, whenever \( x_1 < x_2 \), it follows that \( f(x_1) > f(x_2) \).
This approach sets the foundation for a more structured and logical understanding.
By directly applying the definition, we ensure that the proof is robust and the conclusion is based purely on mathematical logic. The use of direct proof aligns with producing evidence through clear, logical, and established mathematical principles.
This approach sets the foundation for a more structured and logical understanding.
By directly applying the definition, we ensure that the proof is robust and the conclusion is based purely on mathematical logic. The use of direct proof aligns with producing evidence through clear, logical, and established mathematical principles.
Function Behavior
Understanding the behavior of a function like \( f(x) = \frac{1}{x} \) involves more than just computing its output for different inputs.We need to analyze how the function responds as its input changes, which is crucial for determining aspects such as whether the function increases or decreases over specific intervals.
The function behavior is significantly influenced by the value of \( x \): for \( f(x) = \frac{1}{x} \), as \( x \) increases, \( f(x) \) decreases. This occurs because the reciprocal grows smaller with larger \( x \).
Recognizing patterns in function behavior over intervals helps in constructing meaningful interpretations and predictions, which are fundamental for applications in real-world scenarios and advanced mathematics.
The function behavior is significantly influenced by the value of \( x \): for \( f(x) = \frac{1}{x} \), as \( x \) increases, \( f(x) \) decreases. This occurs because the reciprocal grows smaller with larger \( x \).
Recognizing patterns in function behavior over intervals helps in constructing meaningful interpretations and predictions, which are fundamental for applications in real-world scenarios and advanced mathematics.
Interval Analysis
Interval analysis involves examining the behavior of functions over specific ranges or intervals on the number line.In our example with \( f(x) = \frac{1}{x} \), we're interested in the interval \((0, \infty)\).
For this function, because \( \frac{1}{x} \) yields a positive result when \( x > 0 \) and decreases as \( x \) increases, the function is shown to decrease over the interval \((0, \infty)\).
Interval analysis is a powerful method to understand where functions are increasing or decreasing, allowing us to make accurate conclusions for long-term trends and behaviors. This technique is particularly useful when solving calculus problems and in studying continuous function behaviors across different domains.
For this function, because \( \frac{1}{x} \) yields a positive result when \( x > 0 \) and decreases as \( x \) increases, the function is shown to decrease over the interval \((0, \infty)\).
Interval analysis is a powerful method to understand where functions are increasing or decreasing, allowing us to make accurate conclusions for long-term trends and behaviors. This technique is particularly useful when solving calculus problems and in studying continuous function behaviors across different domains.
Other exercises in this chapter
Problem 99
Proof Use the definitions of increasing and decreasing functions to prove that \(f(x)=x^{3}\) is increasing on \((-\infty, \infty) .\)
View solution Problem 100
Distance \(A\) line with slope \(m\) passes through the point \((0,-2) .\) (a) Write the shortest distance \(d\) between the line and the point \((4,2)\) as a f
View solution Problem 101
Proof Prove that if $$p(x)=a_{n} x^{n}+\cdots+a_{1} x+a_{0}$$ and $$q(x)=b_{m} x^{m}+\cdots+b_{1} x+b_{0}$$ where \(a_{n} \neq 0\) and \(b_{m} \neq 0,\) then $$
View solution Problem 102
Proof Use the definition of infinite limits at infinity to prove that \(\lim _{x \rightarrow \infty} x^{3}=\infty\)
View solution