Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ g(x)=\left\lfloor\frac{x}{2}\right\rfloor $$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understanding Even and Odd Functions

A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \), and it is odd if \( f(-x) = -f(x) \) for all \( x \). If neither of these conditions is satisfied, the function is neither even nor odd.

Step 2: Determine \( g(-x) \)

Calculate \( g(-x) \) for the given function. If \( g(x) = \left\lfloor\frac{x}{2}\right\rfloor \), then \( g(-x) = \left\lfloor\frac{-x}{2}\right\rfloor \).

Step 3: Test for Evenness

To test if \( g(x) \) is even, check whether \( g(-x) = g(x) \). Calculate both \( \left\lfloor\frac{x}{2}\right\rfloor \) and \( \left\lfloor\frac{-x}{2}\right\rfloor \). For a specific example, if \( x = 3 \), \( g(x) = 1 \) and \( g(-x) = -2 \). Therefore, \( g(-x) eq g(x) \). Thus, \( g(x) \) is not even.

Step 4: Test for Oddness

To test if \( g(x) \) is odd, check whether \( g(-x) = -g(x) \). Using the same example, \( g(-3) = -2 \) and \(-g(3) = -1 \). Since \( g(-x) eq -g(x) \), \( g(x) \) is not odd.

Step 5: Conclude Function Type

Since \( g(x) \) is neither even nor odd, the function does not satisfy the conditions for either type. Therefore, the function \( g(x) \) is neither even nor odd.

Step 6: Sketch the Graph

Plot the function \( g(x) = \left\lfloor\frac{x}{2}\right\rfloor \). This is a step function that decreases by 1 unit at each odd integer \( x \). The graph will show horizontal segments with steps at each \( x = 2n + 1 \), where \( n \) is an integer, starting at \( x = 0 \).

Key Concepts

even and odd functions

When dealing with functions, it's important to understand whether they are even, odd, or neither. These classifications can help you understand how the function behaves visually, particularly regarding symmetry.

**Even Functions**
- An even function is symmetric about the y-axis.
- Mathematically, a function is even if for every value of \( x \), \( f(-x) = f(x) \).
- A classic example is the function \( f(x) = x^2 \), which mirrors itself on the y-axis.

**Odd Functions**
- An odd function has rotational symmetry about the origin, meaning if you rotate the graph 180 degrees around the origin, it looks the same.
- For a function to be odd, it must satisfy the condition \( f(-x) = -f(x) \).
- Examples include \( f(x) = x^3 \) or \( f(x) = \sin(x) \).

If neither of these conditions is met, the function is neither even nor odd, just like our function \( g(x)=\left\lfloor\frac{x}{2}\right\rfloor \), which neither reflects across the y-axis nor has rotational symmetry at the origin.

function symmetry

Function symmetry is a crucial concept in mathematics, revealing how a function behaves when its inputs are transformed. This concept allows us to predict and plot graphs more easily.

**Y-Axis Symmetry (Even Functions)**
- This symmetry shows if the graph remains unchanged when flipped horizontally along the y-axis.
- The condition \( f(-x) = f(x) \) signifies this symmetry, like in cosine functions.
- If a function's graph can be folded along the y-axis and the two halves match perfectly, it is even.

**Origin Symmetry (Odd Functions)**
- A function has origin symmetry if flipping it over the origin results in no change, akin to rotation.- This symmetry is expressed as \( f(-x) = -f(x) \).
- The sine function is a common example possessing origin symmetry.

Visualizing these symmetries in step functions, like \( g(x)=\left\lfloor\frac{x}{2}\right\rfloor \), helps identify the nature of the graph. Step functions often break these typical symmetry patterns due to their piecewise nature.

piecewise functions

Piecewise functions are powerful mathematical tools that allow different expressions for different intervals of the input. They are essential for modeling scenarios where a single formula is insufficient to describe all behaviors.

**Defining Piecewise Functions**
- Piecewise functions, as the name suggests, involve multiple "pieces" or segments.
- Each piece is defined over a particular interval, separated by specific conditions or steps.
- The step function \( g(x) = \left\lfloor\frac{x}{2}\right\rfloor \) is an example. It uses the floor function, which rounds down to the nearest integer.

**Graphing Piecewise Functions**
- These functions typically display distinct graphical features due to changes in behavior at the interval boundaries.
- The graph may appear "disjointed," as it has different segments rising, falling, or remaining constant, but remains an important depiction.
- In plotting \( g(x) \), there's a drop of one unit for each step as \( x \) crosses an odd integer.

Thus, when working with piecewise functions, it’s crucial to consider each segment's definition and the corresponding interval to understand the overall function's behavior.