Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ G(x)=[2 x-1] $$

Step-by-Step Solution

Answer

The function is neither even nor odd. Its graph is a straight line with a slope of 2 and a y-intercept of -1.

Step 1: Identify Even and Odd Function Criteria

A function is even if \( f(-x) = f(x) \) for all \( x \) in its domain. It is odd if \( f(-x) = -f(x) \). If neither condition is satisfied, the function is neither even nor odd.

Step 2: Calculate \( G(-x) \)

Substitute \(-x\) into the function: \[ G(-x) = [2(-x) - 1] \] Simplifying gives: \[ G(-x) = [-2x - 1] \]

Step 3: Compare \( G(x) \) and \( G(-x) \)

The original function is \( G(x) = [2x - 1] \). The function \( G(-x) = [-2x - 1] \) is neither equal to \( G(x) \) nor \( -G(x) \). Therefore, the function is neither even nor odd.

Step 4: Sketch the Graph

To sketch the graph of \( G(x) = [2x - 1] \), note that \([2x - 1]\) is a linear function with a slope of 2 and a y-intercept of -1. Plot the point (0, -1) and use the slope to plot another point, such as (1, 1). Draw a straight line through these points.

Key Concepts

Understanding Function Properties: Even and Odd Functions

Functions, in mathematics, display unique properties that help us identify their behavior. One key attribute involves categorizing functions as 'even' or 'odd'. These classifications are crucial in simplifying complex mathematical problems, because they aid in understanding the symmetry of functions.
- **Even Functions**: An even function is symmetric about the y-axis. The mathematical definition is that for a function \( f(x) \), if \( f(-x) = f(x) \) holds true for every \( x \) in the domain, then the function is even.- **Odd Functions**: An odd function has symmetry about the origin. A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain.
Understanding these properties aids us in predicting the graph's shape and interpreting the function efficiently.

Graph Sketching: Bringing Functions to Life

Graph sketching is an invaluable skill in visualizing and interpreting mathematical functions. For linear functions like \( G(x) = [2x-1] \), sketching involves several straightforward steps.
First, identify the linear equation form, here it's \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. For \( G(x) \), the slope \( m \) is 2 and the y-intercept \( c \) is -1.
- **Start with the Y-Intercept**: Begin by plotting the y-intercept on the graph. For this equation, plot the point \( (0, -1) \).- **Utilize the Slope**: From the y-intercept, use the slope to find another point on the line. The slope of 2 indicates that for every increase of 1 unit in \( x \), \( y \) increases by 2 units. Thus, moving from \( (0, -1) \) to \( (1, 1) \).- **Draw the Line**: Connect these points with a straight edge to extend the line.
This simple two-step process helps in sketching precise graphs of linear functions, allowing easy further analysis.

Linear Functions: Basics and Characteristics

Linear functions form the backbone of high school algebra. They are vital due to their straightforward relationship between variables, characterized by constant rates of change. A linear function can be expressed in the form \( y = mx + c \), where:- **\( m \)** is the slope: It represents the rate of change of the function. In real-world terms, it indicates how fast or slow one variable changes in relation to another.
- **\( c \)** is the y-intercept: This is the value where the function crosses the y-axis, essentially the starting point when \( x = 0 \).Understanding linear functions helps in modeling situations with a constant rate of change, such as distance over time or cost per item. Furthermore, recognizing their graph as a straight line with defined slope and intercept aids in solving a variety of mathematical problems.