Problem 31 - Algebra Form and Function

Question

In Exercises \(31-36, a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(x)=\frac{x^{3}}{3} $$

Step-by-Step Solution

Answer

Question: Given the function \(f(x) = \frac{x^3}{3}\), the points \(a\) and \(b\) are positive numbers with \(a > b\). Which of the following is true: a) f(a) > f(b) b) f(a) < f(b) c) f(a) = f(b) Answer: a) f(a) > f(b)

Step 1: Find the function values for 'a' and 'b'

First, let's find the value of the function \(f(x)\) at the given points \(a\) and \(b\) by plugging in the values of \(a\) and \(b\) into the function. $$ f(a) = \frac{a^3}{3} $$ $$ f(b) = \frac{b^3}{3} $$

Step 2: Compare the Function Values

Now, we need to compare these function values to see which is larger. Since both \(a\) and \(b\) are positive and \(a > b\), squaring or cubing a positive number will always give a positive result, and since \(a^3 > b^3\), it also means that the following inequality holds: $$ \frac{a^3}{3} > \frac{b^3}{3} $$

Step 3: Conclusion

Since we found that \(\frac{a^3}{3}\) is greater than \(\frac{b^3}{3}\), it follows that \(f(a)\) is larger than \(f(b)\).

Key Concepts

Positive Numbers

Positive numbers are numbers greater than zero. They do not have a negative sign and are found to the right of zero on a number line. When dealing with positive numbers in algebra or calculus, these numbers have some interesting properties that often simplify calculations.
For instance, when you multiply or divide a positive number by another positive number, the result remains positive. This consistency serves as a foundation for many mathematical operations and simplifies the comparison of function values, such as in this exercise. With functions like \( f(x) = \frac{x^3}{3} \), if \( a \) and \( b \) are positive, then both \( f(a) \) and \( f(b) \) will also yield positive results.

Inequality

Inequalities are used to compare the relative size of two numbers or expressions. In this exercise, we are interested in knowing which is larger between \( f(a) \) and \( f(b) \). Given that \( a > b \) and both \( a \) and \( b \) are positive numbers, we can apply an inequality to the function.
Since \( a^3 > b^3 \) generally holds true when \( a > b \), dividing both expressions by a positive constant (in this case, 3) does not change the direction of the inequality. Thus, we conclude that \( f(a) = \frac{a^3}{3} \) is greater than \( f(b) = \frac{b^3}{3} \). Inequalities help us make these comparisons straightforward and logical without needing specific numerical values.

Cubing a Number

Cubing a number means multiplying it by itself twice: \( x^3 = x \times x \times x \). For positive numbers, cubing maintains the positive sign and magnifies the size of the number.
This property is critical in our exercise: since \( a > b \) and both are positive, \( a^3 \) will certainly be greater than \( b^3 \).

• The cube of a larger number is itself larger than the cube of a smaller number, given both are positive.
• The spread between cubes increases as the base numbers increase.

Thus, when comparing \( a^3 \) and \( b^3 \), the inequality \( a^3 > b^3 \) is straightforward to understand.

Function Values

A function assigns a single value to each number within its domain. Here, the function \( f(x) = \frac{x^3}{3} \) provides us with such values. It uses the operation of cubing a number and then scaling it by dividing by 3.
To compare \( f(a) \) and \( f(b) \), you substitute \( a \) and \( b \) into the function. This substitution gives us \( f(a) = \frac{a^3}{3} \) and \( f(b) = \frac{b^3}{3} \). Since we already established that \( a^3 > b^3 \), dividing both by the same positive number 3 does not alter their relative size.