Problem 66 - Algebra 2

Question

Identify each function as S for step, C for constant, A for absolute value, or P for piecewise. \(f(x)=\left\\{\begin{array}{c}{1 \text { if } x>0} \\ {-1 \text { if } x \leq 0}\end{array}\right.\)

Step-by-Step Solution

Answer

The function is piecewise (P).

Step 1: Understand function type definitions

Before identifying the function, let's understand the definitions. Step functions have distinct intervals with constant values. Constant functions have the same output for every input. Absolute value functions have outputs that depend only on the size, not the sign, of the input. Piecewise functions are defined by different expressions over different intervals.

Step 2: Analyze given function

The given function is defined as: \[f(x) = \begin{cases} 1 & \text{if } x > 0 \-1 & \text{if } x \leq 0\end{cases}\] This means that the function uses two different constant values for distinct intervals of "x".

Step 3: Identify function based on description

Given that the function has distinct constant values over different intervals, it follows the characteristic of a piecewise function. Piecewise functions have different rules depending on the input domain.

Key Concepts

step function

Step functions are intriguing because they look like a staircase graphically—hence, their name. A step function maintains constant values across specific intervals but can abruptly change at certain points, much like taking steps up or down. This unique behavior means that the function is not continuous across the entire domain.

Step functions are defined mathematically by different constant values assigned to different sections or intervals of the input variable, often denoted as "x". For example, the given function:

• if \(x > 0\), then \(f(x) = 1\).
• if \(x \leq 0\), then \(f(x) = -1\).

This description highlights the step function’s core property: its discontinuity at specific points, in this case, where \(x = 0\).

In real-life scenarios, step functions might be used in situations where decisions don't change gradually but rather suddenly, such as in tiered pricing or tax brackets.

constant function

A constant function is characterized by the simplest possible rule: it produces the same output regardless of the input. This means its graph is just a horizontal line across the plane, showing no change as "x" varies.

Mathematically, a constant function is written as \(f(x) = c\), where \(c\) is a constant value. Every value of "x" yields the result \(c\).

The key feature of constant functions is that they are both continuous and straightforward, making them an elementary example of mathematical functions. They’re particularly useful in basic modeling scenarios where a fixed result is required, such as maintaining a steady state condition.

function intervals

Function intervals are segments over which a function maintains certain characteristics or behaviors. They are essential when working with piecewise functions, as they divide the function’s domain into parts, each with a specific rule or expression.

For instance, consider our function:

• The interval \(x > 0\) is associated with the value 1.
• The interval \(x \leq 0\) is linked to the value -1.

Each interval tells us exactly how the function behaves over that range. Intervals are not just limited to piecewise functions; they are a critical part of understanding any function’s domain and range, helping define where certain properties hold true.

Understanding function intervals allow us to predict and graph the function's behavior conveniently. In mathematics, clearly defining these intervals is crucial for effectively analyzing and working with any functional expression.