Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ F(x)=3 x-\sqrt{2} $$

Step-by-Step Solution

Answer

The function is neither even nor odd, and its graph is a straight line with slope 3 and y-intercept \(-\sqrt{2}\).

Step 1: Understanding Even and Odd Functions

An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \) in the function's domain. An odd function satisfies the condition \( f(-x) = -f(x) \). If neither condition is met, the function is neither even nor odd.

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

Substitute \( -x \) in place of \( x \) in the function \( F(x) = 3x - \sqrt{2} \). This gives \( F(-x) = 3(-x) - \sqrt{2} = -3x - \sqrt{2} \).

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

Compare \( F(x) = 3x - \sqrt{2} \) with \( F(-x) = -3x - \sqrt{2} \):- Since \( F(-x) eq F(x) \), it is not even.- Since \( F(-x) eq -F(x) \), it is not odd.

Step 4: Conclusion on Even or Odd Nature

Since \( F(-x) eq F(x) \) and \( F(-x) eq -F(x) \), the function \( F(x) = 3x - \sqrt{2} \) is neither even nor odd.

Step 5: Sketching the Graph

The function \( F(x) = 3x - \sqrt{2} \) is a linear equation, forming a straight line with a slope of 3 and a y-intercept at \( -\sqrt{2} \). To sketch:1. Plot the y-intercept at \( (0, -\sqrt{2}) \).2. Use the slope to plot another point. Starting from the y-intercept, go up 3 units and right 1 unit to approximate another point, such as \( (1, 3-\sqrt{2}) \).3. Draw a straight line through the points.

Key Concepts

Linear Functions

Linear functions are the simplest type of functions that create a straight line when graphed. They are defined by the equation of the form \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept of the line. In our given function \( F(x) = 3x - \sqrt{2} \), it follows this pattern. Here, the slope \( m \) is 3 and the y-intercept \( b \) is \(-\sqrt{2}\). The slope indicates how steep the line is and the direction it tilts. A positive slope means the line rises from left to right, while a negative slope means it falls. The y-intercept is the point where the line crosses the y-axis.

Graph Sketching

Graph sketching for linear functions involves plotting the y-intercept and then using the slope to determine another point on the line.
For the function \( F(x) = 3x - \sqrt{2} \), we start by plotting the y-intercept, \( (0, -\sqrt{2}) \). From this point, the slope tells us to move up 3 units and right 1 unit to find the next point, which is approximately \( (1, 3-\sqrt{2}) \).
After plotting these points, we draw a straight line through them. This line represents the graph of the function. It’s important to ensure the line is consistent through the points plotted, confirming the accuracy of your graph.

Y-Intercept

The y-intercept is a crucial component of linear functions. It indicates where the line crosses the y-axis. In the equation \( y = mx + b \), \( b \) is the y-intercept. For our function \( F(x) = 3x - \sqrt{2} \), the y-intercept is \(-\sqrt{2}\).
This intercept provides a starting point for graphing the function. By plotting the y-intercept first, you establish a clear reference point on the graph. This point is then connected to other points defined by the slope, forming the complete line.
Understanding and accurately plotting the y-intercept ensures precision, which is particularly beneficial when dealing with real-world data represented by linear functions.

Function Symmetry

Function symmetry helps determine if a function is even, odd, or neither.
An even function is symmetrical about the y-axis and satisfies \( f(-x) = f(x) \). An odd function has rotational symmetry around the origin and satisfies \( f(-x) = -f(x) \). If neither condition is met, the function lacks symmetry in these respects.
In our case, for \( F(x) = 3x - \sqrt{2} \), substituting \(-x\) gives \( F(-x) = -3x - \sqrt{2} \). Since this does not equal \( F(x) \) or \(-F(x) \), the function is neither even nor odd. This understanding of symmetry is essential as it helps predict the behavior of the function visually, aiding in sketching and analyzing the graph.