Problem 68 - A Graphical Approach to College Algebra

Question

If \(f(x)\) defines a constant function over \((-\infty, \infty),\) how many elements are in the range of \((f \circ f)(x) ?\)

Step-by-Step Solution

Answer

There is 1 element in the range of \((f \circ f)(x)\).

Step 1: Understanding Constant Functions

A constant function is a function that always returns the same value no matter what the input is. If \( f(x) = c \), where \(c\) is a constant, the range of \(f(x)\) is \( \{c\} \), meaning it has only one element.

Step 2: Defining the Composition Function

The composition of a function \(f\) with itself, denoted \((f \circ f)(x)\), means applying \(f\) to the result of \(f(x)\). Given \( f(x) = c \), applying \(f\) again results in \( f(c) \), which is still \(c\). Therefore, \( (f \circ f)(x) = c \).

Step 3: Determine the Range of the Composite Function

Since \((f \circ f)(x) = c\) for all \(x\), the range of \((f \circ f)(x)\) is the set containing only \(c\). Thus, there is only one element in the range, \( \{c\} \).

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics, allowing us to build new functions by combining existing ones. When you compose two functions, you're essentially feeding the output of one function directly into another function. Let's break this down a bit more.
Suppose you have two functions, say \( f(x) \) and \( g(x) \). The composition of these functions, denoted by \((f \circ g)(x)\), means you first apply \(g(x)\), and then take the result to apply \(f\) on it.

• Order matters: When composing functions, the order of operation is important - \((f \circ g)(x)\) is not the same as \((g \circ f)(x)\) unless \(f(x)\) and \(g(x)\) are specifically designed functions like constant functions.
• Notation: The notation \((f \circ g)(x)\) is shorthand for \(f(g(x))\). This tells us to apply \(g\) first, then \(f\).
• Example: If \(g(x) = 2x\) and \(f(x) = x + 3\), then \((f \circ g)(x) = f(2x)\) which simplifies to \(2x + 3\).

The absence of variability in \(f(x) = c\) makes \((f \circ f)(x)\) remain a constant, as the value never changes beyond \(c\). This forms a continuous unraveling of logic connecting two functions.

Range of a Function

The range of a function is the set of all possible output values it can produce. It is essential in understanding the behavior and scope of a function, giving insight into what values the function can take under different input scenarios.
For a constant function like \(f(x) = c\), the range is quite straightforward. Since every input \(x\) yields the output \(c\), the range is simply \(\{c\}\), a set with a single element.

• Constant function range: For a function \(f(x) = c\), regardless of what \(x\) is, \(f(x)\) always equals \(c\). Thus, the range is always \(\{c\}\).
• Range in terms of composite functions: When considering the range of a composite function like \((f \circ f)(x)\), the range remains constrained by the constant values achieved by \(f\).
• General understanding: For functions that are not constant, the range can be more diverse and is influenced by the nature of the function itself.

Understanding the range of a function allows us to make predictions about its behavior, whether we're looking at simple constant scenarios or more complex variable ones.

Composite Function

Composite functions are a marvel of mathematical design, letting us encapsulate more complex behaviors by nesting functions within one another. This method can simplify problem-solving by reducing large steps into smaller, more manageable processes.
Let's delve into how this applies to a constant function like \(f(x) = c\). When you compose \(f\) with itself, we get \((f \circ f)(x)\). Since \(f(x)\) outputs a constant \(c\) for any \(x\), the composite function \((f \circ f)(x)\) mirrors this behavior. The result is also a constant \(c\), showing that the compositional layers do not add complexity in this case.

• Understanding \((f \circ f)(x)\): Here \((f \circ f)(x) = f(f(x)) = f(c) = c\). This continues from the fact that the output of \(f(x)\) provides no new variables.
• Simplicity in constancy: Even when functions are composed, if they are constant, the overall behavior remains unchanged. Hence, the range becomes a singular element, \(\{c\}\).
• Wider applications: While constant compositions are simple, in variable functions, composite structures can merge simplicity with complexity.

The principle of a composite function helps in condensing processes while maintaining clarity over repetitive or singular results.