Problem 3 - Algebra 2

Question

For each pair of functions, find \(f \circ g\) and \(g \circ f,\) if they exist. $$ \begin{array}{l}{f=\\{(-1,9),(4,7)\\}} \\\ {g=\\{(-5,4),(7,12),(4,-1)\\}}\end{array} $$

Step-by-Step Solution

Answer

Only \(g \circ f(4) = 12\) exists; \(f \circ g\) does not exist.

Step 1: Understanding Function Composition

Function composition involves applying one function to the results of another. Specifically, the composition \(f \circ g(x)\) means you apply function \(g\) to \(x\) first, and then apply function \(f\) to the result from \(g\). Similarly, \(g \circ f(x)\) involves applying function \(f\) first and then applying function \(g\) to the result.

Step 2: Identifying the Outputs for g

For the function \(g\), we have three pairs: \((-5,4), (7,12), \text{and} (4,-1)\). This means \(g(-5) = 4\), \(g(7) = 12\), and \(g(4) = -1\). Notice that the set \(f\) only applies to outputs 9 and 7.

Step 3: Determine f(g(x))

For \(f \circ g\) to exist, \(f\) must have as inputs the results of each of the outputs of \(g\). In this case, the outputs of \(g\) are 4, 12, and -1. However, none of these are valid inputs for \(f\), which only accepts the values -1 and 4 as inputs. Hence, \(f \circ g\) does not exist.

Step 4: Identify the Outputs for f

For the function \(f\), we have two pairs: \((-1,9), (4,7)\). This means \(f(-1) = 9\) and \(f(4) = 7\).

Step 5: Determine g(f(x))

For \(g \circ f\) to exist, we must apply \(f\) first and then apply \(g\). Applying \(f\) to inputs -1 and 4, we get outputs of 9 and 7, respectively. \(g\) can interpret the value 7, as \(g(7) = 12\). However, \(g\) does not have 9 as an input, so \(g(9)\) does not exist. Hence, \(g \circ f(-1)\) does not exist, but \(g \circ f(4)\) = 12.

Key Concepts

Functions

Functions are the building blocks of algebra. They link inputs to outputs through a rule. When we talk about a function, we mean a special kind of relation where each input corresponds to exactly one output. Think of it like a vending machine: you press a button (input), and you get a snack (output) based on your choice.

Every function has:

• An input, known as the domain.
• An output, known as the range.

In mathematical terms, if you have a function \( f \) and you input \( x \), the output is represented as \( f(x) \). This output depends on the rule defined by \( f \). For instance, for a function \( f \) given by \{(-1, 9), (4, 7)\}, you plug in -1 to get 9, or 4 to get 7.

Function Pairs

Function pairs are like pairs of shoes; they work together to give specific input-output results. These sets of pairs can be thought of as ordered pairs \((x, y)\) where each \(x\) corresponds to \(y\) following the function's rule.

When we observe a function like \( g \) defined by pairs \((-5, 4), (7, 12), (4, -1)\), this indicates:

• If you start with -5, you'll end up with 4.
• If you choose 7, you'll get 12.

These pairs tell us the exact transformation an input goes through to become an output. Understanding these pairs is crucial when dealing with function compositions because you need to anticipate both sides of the operation.

Inputs and Outputs

Inputs and outputs are the bread and butter of functions and function compositions. They are the starting and ending points of any function's journey.

Inputs are the values you provide to a function, while outputs are the results you get after applying the function's rule. For instance, in the pair \((-1, 9)\) for function \( f \), -1 is the input and 9 is the output. This straightforward process explains the function's purpose - converting inputs to outputs as per its defining pairs.

When tackling function compositions, it’s vital to ensure that the output of one function can be an input for another. For example, for \( f \) and \( g \) to form a valid function composition \( f \circ g \), the outputs of \( g \) must align with the inputs allowed by \( f \). If there’s no match, as seen in this exercise where the outputs of \( g \) (4, 12, -1) don't align with the inputs expected by \( f \) (which are -1 and 4), the composition cannot exist.