Problem 23 - Algebra 2

Question

For each pair of functions, find \(f \circ g\) and \(g \circ f,\) if they exist. $$ \begin{array}{l}{f=\\{(1,2),(3,4),(5,4)\\}} \\\ {g=\\{(2,5),(4,3)\\}}\end{array} $$

Step-by-Step Solution

Answer

\( f \circ g = \{(4,4)\} \), \( g \circ f = \{(1,5), (3,3), (5,3)\}.\)

Step 1: Understanding the Composition

The composition of two functions, written as \( f \circ g \) or \( g \circ f \), involves applying one function to the result of the other. For \( f \circ g \), you first apply \( g \) and then \( f \). For \( g \circ f \), you first apply \( f \) and then \( g \). The composition only exists when the range of the first function coincides with the domain of the second function.

Step 2: Checking f ∘ g

To find \( f \circ g \), apply \( g \) first and then \( f \). Check if the outputs of \( g \) match any inputs of \( f \).- \( g(2) = 5 \) is not in the domain of \( f \).- \( g(4) = 3 \) which is in the domain of \( f \) and \( f(3) = 4 \).Thus, \( f \circ g = \{(4,4)\} \).

Step 3: Checking g ∘ f

To find \( g \circ f \), apply \( f \) first and then \( g \). Check if the outputs of \( f \) match any inputs of \( g \).- \( f(1) = 2 \) and \( g(2) = 5 \) so \( (1,5) \).- \( f(3) = 4 \) and \( g(4) = 3 \) so \( (3,3) \).- \( f(5) = 4 \) and \( g(4) = 3 \) so \( (5,3) \).Thus, \( g \circ f = \{(1,5), (3,3), (5,3)\} \).

Key Concepts

Composite Functions

Composite functions involve combining two functions to create a new one. This is done by using the output of one function as the input for the second function. The notation for a composite function is either \( f \circ g \) or \( g \circ f \), depending on the order in which you apply the functions. For \( f \circ g \), start by applying function \( g \) to your input and use the result as the input for function \( f \). Conversely, \( g \circ f \) means you start with \( f \) and then apply \( g \) .

• Example: If your functions are given as \( f = \{(1,2),(3,4),(5,4)\} \) and \( g = \{(2,5),(4,3)\} \).
• To find \( f \circ g \): Apply \( g \) first. If \( g(x) \) matches any \( f \)'s domain values, only then \( f(x) \) can be calculated.
• To find \( g \circ f \): Apply \( f \) first and then \( g \) to the results.

Understanding these steps ensures you compose the functions accurately.

Domain and Range

In functions, the domain refers to all possible input values, while the range refers to all possible output values. When dealing with composite functions, ensuring the range of the first function aligns with the domain of the second is crucial.

• For \( f \circ g \): Check if the range of \( g \) has values that are in the domain of \( f \).
• For \( g \circ f \): Ensure the range of \( f \) fits within the domain of \( g \).

This exercise has clear steps, for example:- The range of \( g \) is \{5, 3\}, which needs to match some of \( f \)'s domain values for \( f \circ g \).- Similarly, the range of \( f \) is \{2, 4\}, which should align with some values in \( g\)'s domain for \( g \circ f \).This concept is pivotal because it determines whether a composition is possible.

Mapping Diagrams

Mapping diagrams visually represent how elements from the domain of a function are paired with elements in the range. They are useful tools for understanding function compositions.
Imagine each function, \( f \) and \( g \), as a series of arrows between numbers:

• Draw arrows from each input in \( g \) to outputs, clarifying which outputs fit into \( f\).
• Repeat with \( f \) to \( g \) to understand \( g \circ f \).

Mapping diagrams reveal whether the necessary connections exist:- For \( f \circ g \), it's represented by arrows showing the transformation \( g(4) = 3 \rightarrow f(3) = 4 \).- For \( g \circ f \), start from \( f \)'s input/output pairs leading into \( g \), such as \( (1,2) \rightarrow (2,5) \).These diagrams are a visual aid to verify the calculated compositions and enhance comprehension of function connectivity.