Problem 63 - Precalculus : Building Concepts and Connections

Question

Evaluate the given quantity by referring to the function \(f\) given in the following table. $$\begin{array}{cc}x & f(x) \\\\-2 & 1 \\\\-1 & 2 \\\0 & 0 \\\1 & -1 \\\2 & -2\end{array}$$ $$f^{-1}(1)$$

Step-by-Step Solution

Answer

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

Step 1: Understand the meaning of the inverse function

The inverse of a function \(f\), denoted by \(f^{-1}\), is a function that 'reverses' the effect of \(f\). In this case, we are asked to find \(f^{-1}(1)\), which means we have to find the value of \(x\) for which \(f(x) = 1\)

Step 2: Refer to the function table

To find \(f^{-1}(1)\), we need to see at what value of \(x\), the function \(f\) is equal to 1. According to the given table, \(f(-2) = 1\)

Step 3: Conclude from the table

Since \(f(-2) = 1\), it means that the inverse function \(f^{-1}(1)\) is equal to -2. In other words, applying \(f^{-1}\) to 1 leads us back to -2.

Key Concepts

Inverse of a Function

Understanding the inverse of a function is like learning how to retrace your steps in a maze. When we have a function, denoted as f, it's a rule that takes an input and produces an output. Now, imagine you want to go backward: if you have the output, what input did generate it? Enter the concept of the inverse function, which we write as f^-1. This smart companion allows us to reverse the journey or, in mathematical terms, if f(x) = y, then f^-1(y) = x.

Exploring the realms of functions, a student's challenge often lies in the 'function switcheroo'. To master the inverse, you need to think backward. If the function takes you from A to B, the inverse takes you back from B to A. So, if a function says 'I will add 3 to your number', its inverse says 'I will take away 3 to find your original number'.

A quintessential trick to finding the inverse is to swap the roles of the x and y, overturning the function's original formula and then resolving it for y to get the formula for f^-1. But, when working with a function table, the inverse is discovered through detective work in the table's entries, looking for where the output hides and revealing its secret input identity.

Function Evaluation

Evaluating a function is like being given a magic box where you drop a number in, and voilà, a new number comes out. The process of function evaluation is applying the rule of the function to a given number, or input, to calculate the result, or output. When we use the term evaluate, we're following the treasure map that the function provides to find the value of f(x) at a particular x.

Let's say you have a function f that gloriously transforms any number by multiplying it by 2. Evaluating this function for the input 3 means you'll do the simple math: f(3) = 2 * 3 = 6. The essence of function evaluation is plugging in the inputs into the function's rule, which could be an equation or even a set of instructions or conditions that need to be followed meticulously.

When assessing your understanding of function evaluation, check whether you're carefully substituting the value for x and then accurately performing the operations indicated by the function. It's a blend of focused attention to detail and following mathematical procedures, leading you to the correct outcome.

Function Table

A function table is like a schedule for a function, explicitly showing the time slots (inputs) and the corresponding events (outputs). It's a two-column chart where one column lists the inputs and the other reveals the outputs after the function has done its job on the inputs. In simpler terms, it's a helpful list that says 'for this x, I get this y'.

Function tables are particularly handy when visualizing how a function behaves without getting tangled up in equations. They paint a clear picture of the relationship between input and output values. For instance, the table for a function that adds 2 to any input would show an increase of 2 in the output column for each entry in the input column. It's a powerful visual aid in understanding and analyzing the nature of a function.

To use a function table effectively, focus on the pattern of change from one row to the next. Understanding the nature of these changes can give insights into the rate of growth or decline, help find the existence of an inverse, and allow quick function evaluations. It's a perfect stepping stone for those practicing the art of recognizing patterns in numbers and translating a function's abstract notion into concrete examples.