Problem 63 - A Graphical Approach to Precalculus with Limits

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

The range of \((f \circ f)(x)\) has 1 element.

Step 1: Understanding the Problem

The function \(f(x)\) is a constant function, which means it has the same output value for any input \(x\) in its domain of \((-\infty, \infty)\). We are tasked with finding the number of elements in the range of \((f \circ f)(x)\), which is the composition of \(f(x)\) with itself.

Step 2: Analyze f(x) as a Constant Function

A constant function can be expressed as \(f(x) = c\), where \(c\) is a constant value. The range of a constant function is simply \{c\}, because it only outputs a single value \(c\) for any \(x\).

Step 3: Determine (f ∘ f)(x)

The composition \((f \circ f)(x)\) means we apply \(f(x)\) to \(f(x)\). Since \(f(x) = c\) as a constant function, \((f \circ f)(x) = f(f(x)) = f(c)\). Because \(f(x)\) is constant, \(f(c) = c\). Therefore, \((f \circ f)(x)\) is also a constant with the value \(c\).

Step 4: Identify the Range of (f ∘ f)(x)

Since \((f \circ f)(x)\) is a constant function with the value \(c\), the range of \((f \circ f)(x)\) is \{c\}. This means it also contains only one unique element.

Step 5: Conclusion

The range of \((f \circ f)(x)\) as a result of the composition is the set \{c\} with just one element. Hence, the number of elements in the range of \((f \circ f)(x)\) is 1.

Key Concepts

Function Composition

Function composition involves creating a new function by applying one function to the results of another. This concept is denoted by \((f \circ g)(x)\), which means applying function \(g(x)\) first and then function \(f(x)\) on the result of \(g(x)\). In simpler terms, it’s like stacking operations where the output of one function becomes the input to the next.

When dealing with a constant function like \(f(x) = c\), function composition can be straightforward. Applying \(f(x)\) to itself, as in \((f \circ f)(x)\), means the constant output \(c\) is used as the input, but since \(f(x) = c\) for any input, the result remains \(c\).

In general, function composition is a powerful mathematical tool that allows us to build more complex operations out of simpler functions. It has various applications in calculus, computer science, and even in solving differential equations.

Range of a Function

The range of a function is the set of all possible output values it can produce. For a constant function \(f(x) = c\), the range is particularly simple. Since a constant function gives the same output no matter what input it takes, its range consists of just one value: \(\{c\}\).

Understanding the range is crucial because it tells us the limitations or constraints of a function’s behavior. For instance, if the range is limited to positive numbers, we know the function never outputs negatives. In our current example, since \((f \circ f)(x)\) is also a constant function with the output value \(c\), its range is \(\{c\}\) as well.

An essential step in solving any problem involving functions is recognizing the range, as it defines the boundaries of what the function can achieve.

Mathematical Problem Solving

Mathematical problem solving is a critical skill that involves using logic and reasoning to find solutions to various math problems. This process can include identifying known information, organizing thoughts, and applying mathematical concepts and techniques.

In our exercise, we began by understanding the nature of the constant function and how function composition works. This insight helped us predict the range of the composed function \((f \circ f)(x)\).

Problem-solving often requires a step-by-step approach:

• Identify the problem and what you need to solve for.
• Understand and analyze the given functions and equations.
• Break down complex problems into simpler parts.
• Apply known mathematical principles to find the solution.
• Verify results to ensure their correctness.

Developing strong problem-solving skills allows us to approach mathematical challenges effectively and can be applied beyond math, in fields like engineering, data science, and everyday decision-making.