Problem 16 - Algebra Form and Function

Question

(a) Is \(y\) proportional, or is it inversely proportional, to a positive power of \(x\) ? (b) Make a table of values showing corresponding values for \(y\) when \(x\) is \(1,10,100,\) and 1000 . (c) Use your table to determine whether \(y\) increases or decreases as \(x\) gets larger. \(y=\frac{1}{x}\)

Step-by-Step Solution

Answer

Short Answer: The function y = 1/x is inversely proportional to x. Based on the table of values, we can conclude that as x increases, y decreases.

Step 1: 1. Determine if y is proportional or inversely proportional to a positive power of x

y = 1/x, which means it is inversely proportional to x because the function is in the form of k/x, with k being a constant. Therefore, y is inversely proportional to x.

Step 2: 2. Create a table of values

We will now create a table of values for y when x is 1, 10, 100, and 1000 using the given function y = 1/x: | x | y | |:-----:|:-----:| | 1 | 1 | | 10 | 0.1 | | 100 | 0.01 | | 1000 | 0.001 |

Step 3: 3. Determine if y increases or decreases as x gets larger

By observing the table, we can see that as x increases, the values of y decrease. Therefore, y decreases as x gets larger.

Key Concepts

Proportionality

Proportionality in mathematics refers to a relationship between two quantities where a change in one quantity results in a proportional change in another. In this context, we have inverse proportionality. When we say that variable \( y \) is inversely proportional to \( x \), it means that as \( x \) increases, \( y \) decreases, and vice versa. This relationship can be mathematically represented as \( y = \frac{k}{x} \), where \( k \) is a constant. In our problem, the equation \( y = \frac{1}{x} \) illustrates inverse proportionality because \( y \) decreases as \( x \) increases. This is a key concept in many real-world applications, such as in understanding how speeds relate to travel times at constant distances.

Power Functions

Power functions are mathematical expressions of the form \( y = ax^n \), where \( a \) is a constant, \( n \) is a real number, and \( x \) is the variable. However, in inverse power functions like \( y = \frac{1}{x^n} \), the presence of \( x \) in the denominator causes different behavior. Here, since the power \( n = 1 \), we directly deal with a basic inverse function \( y = \frac{1}{x} \). Unlike direct power functions, inverse power functions result in decreasing \( y \) as \( x \) grows, which is demonstrated in this problem. As you learn about power functions, it is important to understand how changing the power \( n \) affects the behavior and shape of the graph of the function. Each power describes a distinct kind of rate, or how sharply \( y \) changes with \( x \).

Tables of Values

Tables of values are helpful tools for visualizing and understanding the relationship between variables. They list pairs of corresponding values which help show how one variable changes with respect to another. In our exercise, we constructed a table for \( x \) values of 1, 10, 100, and 1000, and their corresponding \( y \) values based on the equation \( y = \frac{1}{x} \). The table shows:

• When \( x = 1 \), \( y = 1 \)
• When \( x = 10 \), \( y = 0.1 \)
• When \( x = 100 \), \( y = 0.01 \)
• When \( x = 1000 \), \( y = 0.001 \)

By examining tables, you can easily see the trend: as \( x \) increases, \( y \) consistently decreases, confirming the inverse relationship. These tables are invaluable for spotting patterns and behaviors in functions.