Problem 58 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The solution to the composite function \(f^{-1}(g(1))\) is -1.

Step 1: Locate Value for \(g(1)\)

First, locate the x-value in the g table that equals 1. According to the provided table, \(g(1) = 1\).

Step 2: Find Corresponding Value in f table

Now referring to the f table, find the corresponding x-value where f(x) equals the located value i.e., where \(f(x) = g(1) = 1\). According to the provided table, for \(f(-1) = 1\), \(x = -1\).

Step 3: Answer the Expression

So the value of \(f^{-1}(g(1))\) is -1. This is the x-value that makes f(x) = g(1) true, and hence is the solution for the given composite function expression.

Key Concepts

Function Composition

Function composition is a cornerstone concept in mathematics, pivotal in piecing together relationships between different functions. Imagine it as a relay race, where the output of one function becomes the input for another. In simpler terms, given two functions, say, f and g, their composite, denoted as f(g(x)), means you first feed a value into g, take the result, and then plug it into f.

The beauty of function composition lies in its ability to create complex functions from simpler ones. This not only makes problem-solving more efficient but also helps us understand deeper connections within mathematical concepts. When working with tables, like in the exercise, we read from the inside out. First, locate the value of g(1) and then find which x makes f(x) equal to that result - a process similar to following a mapped route, with each table entry as a distinct landmark.

Remember that the order of composition matters. f(g(x)) is generally different from g(f(x)), much like how reversing the order in a relay race would change the outcome. It's always important to follow the correct sequence to reach the desired result.

Inverse Functions

The concept of inverse functions is akin to finding a mathematical 'undo' button. An inverse function, f^{-1}, reverses the effect of the original function, f, meaning that f^{-1}(f(x)) = x and f(f^{-1}(x)) = x. It's like following a path in reverse: if you walked from point A to B following certain steps, the inverse would take you back from B to A using the same steps backward.

In the context of our exercise, when we look for f^{-1}(g(1)), we're searching for the value that, when fed into f, yields g(1). The inverse function effectively traces back to the original x before it was transformed by f. This powerful tool offers a way to decode the function's actions, allowing us to backtrack and reconstruct initial values. For an inverse to exist, each input should be matched to a unique output; that's the rule of the game. In other words, a function must be one-to-one – for every x there's one and only one y, and vice versa – to have an inverse.

Function Evaluation

Function evaluation may seem straightforward, but it's absolutely central to understanding how functions behave. Evaluating a function like f(x) or g(x) involves replacing the variable x with a specific value, and then calculating the result. Think of it as ordering a meal at a restaurant; you choose an item (a specific value), and the kitchen (the function) prepares your order (the output).

In the given exercise, evaluating the composite function involved two evaluations: First, we determined the outcome of g(1), which was like ordering your appetizer. Then, knowing what that appetizer was, we looked to f to prepare the main course, ending up with f^{-1}(g(1)). The process is sequential and systematic, ensuring that each step follows from the last. By mastering function evaluation, one essentially learns to navigate through the function 'menu' with ease, a skill that is essential for tackling more complex mathematical problems. Always ensure you plug the right values into the right function at the right time to avoid any mix-up in the 'orders'.