Problem 97 - College Algebra with Modeling and Visualization

Question

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ \left(g \circ f^{-1}\right)(5) $$

Step-by-Step Solution

Answer

(g \circ f^{-1})(5) = 1.

Step 1: Understand the Composition Formula

The expression \((g \circ f^{-1})(5)\) means we first need to find \(f^{-1}(5)\) (the value of \(x\) such that \(f(x) = 5\)) and then use it as input for \(g(x)\). This means: calculate \(g(f^{-1}(5))\).

Step 2: Identify Value for Inverse Function

From the given table for \(f(x)\), identify the \(x\) value for which \(f(x) = 5\). By observing the table, we see that when \(x = 2\), \(f(x) = 5\). Thus, \(f^{-1}(5) = 2\).

Step 3: Evaluate the Composition

Now use the result from Step 2 to find \(g(2)\). Refer to the table for \(g(x)\) to find the value for \(x = 2\). From the table, \(g(2) = 1\).

Step 4: Conclusion

Since we have found \(g(f^{-1}(5)) = g(2) = 1\), the result of the composition \((g \circ f^{-1})(5)\) is 1.

Key Concepts

Inverse Functions

Understanding inverse functions is crucial when working through problems involving function composition. In simple terms, if you have a function that takes an input and gives an output, the inverse function reverses this operation. For a function like \( f \), the inverse function, denoted as \( f^{-1} \), finds the input that corresponds to a given output. It's essentially flipping the roles of input and output. For instance, if \( f(2) = 5 \), then \( f^{-1}(5) = 2 \). This means by applying \( f \) to 2, you get 5; conversely, applying \( f^{-1} \) to 5 returns you to 2.When it comes to finding the inverse using tables, first look for the function's output value. Then, locate this output in the inverse function to determine the corresponding input. This technique requires care and precision, as each output must match with a unique input.

Evaluating Functions

Evaluating functions means calculating the output for a given input using the function's rule or table of values. This can be likened to looking up a set of instructions to see what result you get when you provide a specific input.Let's consider the function \( g(x) \) from the table:

• To find \( g(2) \), locate 2 in the row of \( x \) values.
• The corresponding output in the table shows \( g(2) = 1 \).

Here, you simply follow the input given (2) to the output column where you find the result.Evaluating functions is straightforward when a table is provided. Each row gives you a direct input-output pair, making the process akin to following a recipe.

Table of Values

A table of values is an organized way of displaying inputs and their corresponding outputs for a function. It helps visualize the relationship between \( x \) values and their function outputs, making it easier to evaluate functions or find inverse functions.To utilize a table effectively:

• Find the row corresponding to the given input or output.
• Read the associated output or input based on what's required, such as when working with inverse functions or compositions.

For example, the table for \( f(x) \) allows you to immediately see that when \( x = 2 \), \( f(x) = 5 \). Similarly, you can scan for \( x = 3 \) to ascertain that \( f(x) = 4 \).Tables make these tasks less daunting by offering a clear, structured view of how one element maps to another. Thus, learning to interpret these tables is a foundational skill in algebra and beyond.