Problem 32 - Algebra Form and Function

Question

\(a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(t)=\frac{\sqrt{5}}{t^{7}} $$

Step-by-Step Solution

Answer

Answer: \(f(b)\) is larger than \(f(a)\).

Step 1: Write down the given function for a and b

Substitute \(a\) and \(b\) into the function \(f(t)\): $$ f(a) = \frac{\sqrt{5}}{a^7} $$ $$ f(b) = \frac{\sqrt{5}}{b^7} $$

Step 2: Compare f(a) and f(b)

To compare \(f(a)\) and \(f(b)\), we can set up a ratio \(\frac{f(b)}{f(a)}\) as follows: $$ \frac{f(b)}{f(a)} = \frac{\frac{\sqrt{5}}{b^7}}{\frac{\sqrt{5}}{a^7}} $$

Step 3: Simplify the ratio

Simplify the ratio by canceling out the common factors of \(\sqrt{5}\) and rewriting the ratio as: $$ \frac{f(b)}{f(a)} = \frac{a^7}{b^7} $$

Step 4: Compare a and b and draw conclusions

We know that \(a > b > 0\). Since both \(a\) and \(b\) are positive and a is larger than b, that means \(a^7 > b^7\). Thus, the ratio \(\frac{f(b)}{f(a)} > 1\). Since \(\frac{f(b)}{f(a)} > 1\), we can conclude that \(f(b)\) is larger than \(f(a)\).

Key Concepts

Exponential Functions

Exponential functions are fundamental mathematical tools characterized by variables in the exponent of a power. For instance, in our exercise, the expression \(t^7\) is an example of an exponential expression where 7 is the exponent. Exponential growth or decay depends on whether the exponent is positive or negative.

These functions are vital in various fields such as finance, biology, and physics because they describe processes that involve rapid changes.

• If the exponent is positive, as in \(a^7\), the function describes growth.
• If the exponent is negative, as in \(t^{-7}\), the function represents decay, which inversely relates to the size of \(t\).

Understanding exponential functions helps in interpreting how varying inputs impact the output of a function significantly. In this case, as the variable \(t\) increases, the value of \(\frac{1}{t^7}\) decreases, illustrating an exponential decay in function \(f(t) = \frac{\sqrt{5}}{t^7}\).

Inequalities

Inequalities are mathematical statements indicating that one value is larger or smaller than another. They help in expressing limits and conditions in mathematical terms. In the exercise, we explored the inequality relationship between \(a\) and \(b\), where \(a > b\).

Understanding inequalities involves:

• Recognizing symbols: \(>\) for greater than, \(<\) for less than.
• Applying operations: Knowing how inequalities change under addition, subtraction, multiplication, or division.

The exercise illustrates that when both numbers are raised to the power of 7, this inequality is preserved, meaning \(a^7 > b^7\). Thus, this understanding is leveraged to determine which of the functions \(f(a)\) or \(f(b)\) is greater depending on the inputs.

Function Evaluation

Function evaluation is crucial in assessing how different inputs affect the outcome of a function. It involves substituting variables with given numerical values to compute the function's result. In this task, \(a\) and \(b\) were plugged into the function \(f(t) = \frac{\sqrt{5}}{t^7}\) to produce \(f(a)\) and \(f(b)\).

To evaluate a function accurately, follow these steps:

• Identify the function's formula and the variable to be substituted.
• Insert the specific values into the function to replace the variable.
• Perform arithmetic operations as dictated by the function's form.

Given that \(a>b\), the evaluation led to \(f(b)\) being greater than \(f(a)\). This approach elucidates how substituting different values can reveal insights into the function’s behavior and comparative relationships.