Problem 98 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

(f^{-1} \circ g)(4) = 2.

Step 1: Evaluate g(4)

To evaluate \( (f^{-1} \circ g)(4) \), we first find \( g(4) \) using the provided table for \( g(x) \). From the table, when \( x = 4 \), \( g(x) = 5 \). Thus, \( g(4) = 5 \).

Step 2: Find the Output of the Composite Function

Now that we know \( g(4) = 5 \), we evaluate \( f^{-1}(5) \). We need to find when \( f(x) = 5 \) from the provided table for \( f(x) \). From the table, \( f(x) = 5 \) when \( x = 2 \). Thus, \( f^{-1}(5) = 2 \).

Step 3: Final Evaluation of the Composite Function

The composite function \( (f^{-1} \circ g)(4) \) can be evaluated as the output of \( f^{-1}(5) \), which we found in the previous step. Hence, \( (f^{-1} \circ g)(4) = f^{-1}(5) = 2 \).

Key Concepts

Inverse Function

When talking about inverse functions, we're essentially discussing how to "undo" a function. If you think of a function as a machine where you input a number and receive a new number as output, an inverse function does the reverse. It takes the output back to its original number.

For example, if you have a function \( f \) such that \( f(x) = 5 \), and you need to find what \( x \) was, you'd use the inverse of \( f \), noted as \( f^{-1} \). The goal is to solve \( f^{-1}(5) = x \). Simply put, an inverse function "reverses" the effect of the original function.

In step-by-step solutions, using the table, you look for the output value in the column related to \( f(x) \), find its corresponding input \( x \) value, and determine that as \( f^{-1} \). This concept is fundamental and often used in composition of functions.

Function Evaluation

Function evaluation involves finding the output of a function given an input. It's akin to consulting a dictionary; you have a word and you look to find its meaning. This process dictates which value a function assigns to a specific input.

To evaluate a function, such as \( g(4) \), simply locate the input \( x \) from the table and determine the corresponding output. In our exercise, the table showed \( g(4) = 5 \). This approach helps to methodically discover outputs without complex calculations.

Evaluating functions is about making sure you accurately interpret tables or equations to find the assigned output. Once you have this value, you can perform further operations such as plugging it into another function or finding an inverse if necessary.

Composite Functions

Composite functions, denoted as \((f \circ g)(x)\), involve applying one function to the result of another. Think of it as using two machines in sequence. You process a number through \( g \), and then take that result and process it through \( f \).

In our example, \((f^{-1} \circ g)(4)\) was calculated by first evaluating \( g(4) \) to obtain 5, then finding the inverse function \( f^{-1}(5)\). The composite operation here is completed by applying two different processes step by step.

Understanding composite functions is key in uncovering how results compound through multiple steps. By evaluating component functions separately and thoughtfully piecing them together, you can solve more complex problems with ease.