Problem 31 - Contemporary Precalculus

Question

Let \(f(t)\) be the population of rabbits on Christy's property \(t\) years after she received 10 of them as a gift. $$\begin{array}{|c|c|}\hline t & f(t) \\\\\hline 0 & 10 \\\\\hline 1 & 23 \\\\\hline 2 & 48 \\\\\hline 3 & 64 \\\\\hline 4 & 70 \\\\\hline 5 & 71 \\\\\hline\end{array}$$ Compute the following, including units, or write "not enough information to tell." \(f^{-1}\) denotes the inverse function of \(f\). (a) \(f(2)\) (b) \(f^{-1}(48)\) (c) \(f^{-1}(71)\) (d) \(3 \cdot f^{-1}(70)\) (e) \(f^{-1}(2 \cdot 48)\) (f) \(f(70)\) (g) \(f^{-1}(4)\)

Step-by-Step Solution

Answer

Answer: 48

Step 1: (a) Calculate \(f(2)\)

To find the value of \(f(2)\), look at the table and find the corresponding output value when the input is \(2\). The input corresponds to the \(t\) column, and the output corresponds to the \(f(t)\) column. We can observe from the table that when \(t = 2\), \(f(t) = 48\). So, \(f(2) = 48\).

Step 2: (b) Calculate \(f^{-1}(48)\)

To find the value of \(f^{-1}(48)\), look at the table and find the corresponding input value when the output is \(48\). The output corresponds to the \(f(t)\) column, and the input corresponds to the \(t\) column. We can observe from the table that when \(f(t) = 48\), \(t = 2\). So, \(f^{-1}(48) = 2\).

Step 3: (c) Calculate \(f^{-1}(71)\)

To find the value of \(f^{-1}(71)\), look at the table and find the corresponding input value when the output is \(71\). The output corresponds to the \(f(t)\) column, and the input corresponds to the \(t\) column. We can observe from the table that when \(f(t) = 71\), \(t = 5\). So, \(f^{-1}(71) = 5\).

Step 4: (d) Calculate \(3 \cdot f^{-1}(70)\)

To calculate this expression, first find the value of \(f^{-1}(70)\). Look at the table and find the corresponding input value when the output is \(70\). The output corresponds to the \(f(t)\) column, and the input corresponds to the \(t\) column. We can observe from the table that when \(f(t) = 70\), \(t = 4\). So, \(f^{-1}(70) = 4\). Now, multiply this value by \(3\): \(3 \cdot f^{-1}(70) = 3 \cdot 4 = 12\).

Step 5: (e) Calculate \(f^{-1}(2 \cdot 48)\)

First, calculate the product \(2 \cdot 48 = 96\). Now, look at the table and try to find the corresponding input value when the output is \(96\). We can see that the value \(96\) is not present in the \(f(t)\) column. Therefore, we do not have enough information to find the value of \(f^{-1}(96)\). So, the answer to this question is "not enough information to tell."

Step 6: (f) Calculate \(f(70)\)

To find the value of \(f(70)\), look at the table and try to find the corresponding output value when the input is \(70\). We can see that the value \(70\) is not present in the \(t\) column. Therefore, we do not have enough information to find the value of \(f(70)\). So, the answer to this question is "not enough information to tell."

Step 7: (g) Calculate \(f^{-1}(4)\)

To find the value of \(f^{-1}(4)\), look at the table and try to find the corresponding input value when the output is \(4\). We can see that the value \(4\) is not present in the \(f(t)\) column. Therefore, we do not have enough information to find the value of \(f^{-1}(4)\). So, the answer to this question is "not enough information to tell."

Key Concepts

Function Evaluation

Evaluating functions is like reading a recipe: you find out what happens when you add specific ingredients. Here, the ingredient is the input value, and the recipe is the function itself. The function notation \( f(t) \) essentially states "what is the value of \( f \)?" when \( t \) is a certain number. To evaluate a function, you simply:

• Find the input in the function's rule or table.
• Look across the row to find the result or output.

For example, to find \( f(2) \) in our given exercise, you look at the table for when \( t = 2 \). You'll then see that the corresponding \( f(t) \) value is \( 48 \). Thus, processing this information easily unfolds the answer: \( f(2) = 48 \). Function evaluation lets you see what output you get from a given input.

Table Analysis

Tables provide a convenient way to summarize and visualize function evaluations. They convert complex decisions into straightforward look-up tasks. In a table summarizing function values, each row displays:

• The input value \( t \) in one column.
• The output value \( f(t) \) in the other column.

Using a table, like the one in Christy's rabbit population example, allows you to quickly analyze relationships. To determine \( f^{-1}(x) \), you're reversing the evaluation process: starting with the output, and finding out what input caused it. So, if you're trying to find \( f^{-1}(71) \), you find the \( 71 \) in the \( f(t) \) column and trace it back to reveal \( t = 5 \). Tables excel in helping us efficiently see patterns and isolate precise values within function relationships.

Rabbit Population Growth

Population growth of rabbits can showcase how a simple model can describe complicated natural phenomena. Understanding function evaluation and table use, like in our rabbit example, helps track these changes over time. With each year as an input, you see corresponding changes in population size. The change might not always be linear or predictable. Rather, it reflects genuine natural complexity, such as

• Exponential growth as resources are initially abundant.
• Plateau or limit as resources thin out (carrying capacity).

This rabbits-on-a-farm scenario is not just about numbers in a table—it can represent real-world biological limits and growth patterns. Knowing how to read and analyze these trends gives insights into ecological dynamics and management challenges for real populations.