Problem 53 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The value of the composite function \(f(g(1))\) is 5.

Step 1: Determine \(g(1)\)

Look in the table for function \(g\) to find the output when \(x=1\). In this case, when \(x=1\) the corresponding output \(g\) is 1, so \(g(1) = 1\).

Step 2: Compute \(f(g(1))\)

Having found that \(g(1)=1\), substitute this into function \(f\) to find \(f(g(1))\). That is, find the output in function f where \(x=1\). According to the function \(f\) table, when \(x=1\), \(f(x)=5\). Therefore, \(f(g(1)) = 5\).

Key Concepts

Function Composition

Understanding function composition is like learning a new language in math that lets us combine two functions to create a new one. It's a process where you take two functions, say f and g, and apply them in succession. The result is noted as f(g(x)), which means you first feed an input x into the function g, then take the output and plug it into the function f.

Picture it like a two-step journey: you start at some point x, walk through function g that transforms it, and then continue through function f that changes it again. This journey ends at the final destination, which is the value of the composite function f(g(x)). It's essential to proceed in the correct order, as swapping f and g can lead to a totally different outcome — just like how taking a wrong turn can lead you to an entirely different place.

Table of Values

A table of values serves as a map that pairs each input with its corresponding output for a function. It’s a neatly organized grid where you can quickly spot what happens to specific values when processed by a function. For example, if you're given an input (or x value), you can simply look across the corresponding row to find the output (or f(x) value).

Using a table of values is a straightforward method to understand functions without needing complex calculations. Think of each row as an individual fact, telling us what the function will definitely output for each given input. Much like looking up a word in a dictionary, a table of values is where we look up an input to find its output.

Substitution Method

The substitution method is like a relay race in math: the baton (an output value from one function) is passed to the next runner (another function) to continue towards the finish line (the final result). This technique involves replacing a variable with its equivalent value. When dealing with composite functions, such as f(g(x)), you first evaluate the innermost function g(x) and get a result. Then, as if we're relaying the baton, we 'substitute' that result into the next function, f, by replacing x in f(x) with the output from g(x). This clarifies the path to finding the value of the composite function.

In practice, when the question asks you to find f(g(1)), you start with finding g(1). This is like the first leg of our relay — determining the value after the first runner. Then, taking that result, we 'hand it off' to the next function, f, to get to the final answer, completing the race.