Problem 57 - Precalculus Essentials

Question

\(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. $$\begin{array}{cccc} \hline x & f(x) & x & g(x) \\ \hline-1 & 1 & -1 & 0 \\ 0 & 4 & 1 & 1 \\ 1 & 5 & 4 & 2 \\ 2 & -1 & 10 & -1 \end{array}$$ $$f^{-1}(g(10))$$

Step-by-Step Solution

Answer

The value of \(f^{-1}(g(10))\) is 2.

Step 1: Find the value of \(g(10)\)

Looking at the table for function \(g\), we can find that \(g(10)\) is -1.

Step 2: Find the inverse of function \(f\)

Since we found that \(g(10)\) is -1, we should find \(f^{-1}(-1)\). The \(f^{-1}(x)\) means that we need to find the \(x\) value such that \(f(x)\) equals -1. Looking at the table for function \(f\), we see that when \(f(x)\) equals -1, the \(x\) value is 2.

Step 3: Evaluate the composite function

Thus, \(f^{-1}(g(10)) = f^{-1}(-1) = 2\).

Key Concepts

Understanding Inverse Functions

Inverse functions are about reversing or undoing the operation of a given function. If we think of a function as a process that transforms an input into an output, an inverse function will transform the output back into the input. To illustrate, if a function \( f \) takes an \( x \) value and gives a result \( y \), then its inverse, denoted as \( f^{-1} \), will take \( y \) and return \( x \). It's like a mathematical version of retracing your steps!

• Notation: The inverse function is usually written as \( f^{-1} \).
• Key Idea: If \( f(a) = b \), then \( f^{-1}(b) = a \).

Inverse functions can only exist for particular types of functions, specifically those that are one-to-one. A one-to-one function means every output is produced by only one specific input, making backtracking possible.

Decoding Function Tables

Function tables are a visual way to map the relationship between inputs and outputs for functions. They work like a chart where you match \( x \) values to their respective \( f(x) \) or \( g(x) \) outputs. Each row in the table shows how a particular input (\( x \) value) is transformed into an output by the function.

• Column Alignment: Often, the first column contains inputs, and the second column lists corresponding outputs.
• Interpreting Data: You use these tables to quickly find values, which is incredibly helpful for evaluating composite functions.
• Reading from Tables: For instance, if you need to find \( f(2) \) and the table shows that \( f(2) = -1 \), then -1 is the output when the input is 2.

Function tables simplify complex calculations, allowing learners to visualize relationships quickly. They are particularly useful for understanding piecewise or discontinuous functions.

Simplifying Function Evaluation

Function evaluation involves finding the output of a function for a given input. It is as simple as plugging in a value for \( x \) and performing any arithmetic or operations dictated by the function. When evaluating composite functions, you might first find the output of one function, then use that result as the input for another function.

• Example: To evaluate \( f(g(x)) \), start by determining \( g(x) \), then use this result to find \( f \).
• Composite Functions: These involve applying multiple functions, such as \( f(g(x)) \) or \( g(f(x)) \), which means you perform one function after the other.

The step-by-step nature of these operations makes them straightforward once you understand how each individual function works. Remember, accuracy is critical here, as each step depends on the previous results.