Problem 63 - Algebra 2

Question

Identify each function as S for step, C for constant, A for absolute value, or P for piecewise. \(h(x)=\frac{2}{3}\)

Step-by-Step Solution

Answer

The function is a Constant function (C).

Step 1: Analyze the Function

Look at the given function expression, which is a simple fraction \(h(x) = \frac{2}{3}\). This format suggests it is a single unchanging value regardless of \(x\).

Step 2: Determine the Function Type

Since the function outputs the same value \(\frac{2}{3}\) for any input \(x\), it indicates that the function is constant. Constant functions are not dependent on the variable \(x\) to change their output value.

Step 3: Classify the Function

Based on the constant nature of the function \(h(x) = \frac{2}{3}\), classify it as a Constant function. It does not exhibit characteristics of a step, absolute value, or piecewise function.

Key Concepts

Constant Function

A constant function is a type of function where the output value remains the same for any input value. In simpler terms, no matter what value you substitute for the variable, the answer will always be a specific number. This makes the graph of a constant function a horizontal line on the Cartesian plane.
For a function like \(h(x) = \frac{2}{3}\), every input \(x\) results in the value \(\frac{2}{3}\). That's because the function does not depend on \(x\) at all. It just stays constant.
Constant functions are usually represented as \(f(x) = c\), where \(c\) is a constant. This means that the function's graph doesn't change direction or have any variation.

• Example: \(f(x) = 5\) is a constant function where every input gives an output of 5.
• Visual Representation: Since \(f(x) = c\) is constant, if you were to plot it, you'd have a straight line parallel to the x-axis crossing the y-axis at \(c\).

Function Classification

Function classification is the process of categorizing functions based on their characteristics and behaviors. It allows us to easily identify the kind of relationship a function exhibits between its input and output. This helps in understanding its properties and potential applications.
Functions are generally classified into several types, including:

• Constant functions, like \(h(x) = \frac{2}{3}\), which have a single value for every input.
• Linear functions, which have a constant rate of change and are represented as straight lines.
• Quadratic functions, which are represented with parabolas and have terms involving squares of variables.
• Piecewise functions that define separate expressions for different sections of their domain.
• Absolute value functions, denoted as \(|x|\) and reflect symmetry around the x-axis.

By understanding these classifications, you can recognize a function's promises and limitations, such as predicting its graph shape or behavior.

Function Analysis

Function analysis involves examining a function to understand its features, including how it behaves with different inputs and affects its output. Analyzing a function is crucial for solving complex mathematical problems and for practical applications in engineering and sciences.
When analyzing a function, consider these key points:

• Domain: All possible input values (\(x\)) a function can accept.
• Range: All possible output values a function can produce.
• Behavior: This encompasses how the function increases, decreases, or remains constant as \(x\) varies.
• Graphical Depiction: Plotting the function on a graph to visualize these relationships.

For a constant function like \(h(x) = \frac{2}{3}\), the domain is usually all real numbers, and the range is solely \(\{\frac{2}{3}\}\), since the output does not vary. The horizontal line on a graph indicates complete stability across the entire domain.