Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(F(x)=2 x+1\)

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Identifying the Type of Function

To determine if the function is even, odd, or neither, let's use the definitions: - A function is **even** if it satisfies the condition \(f(-x) = f(x)\) for all \(x\) in the domain.- A function is **odd** if it satisfies \(f(-x) = -f(x)\) for all \(x\) in the domain. First, we compute \(F(-x)\) for the given function \(F(x) = 2x + 1\).

Step 2: Calculate F(-x)

Substitute \(-x\) into the function. Calculate \(F(-x)\):\[F(-x) = 2(-x) + 1 = -2x + 1.\]

Step 3: Test for Even Function

Compare \(F(-x)\) with \(F(x)\):\[F(x) = 2x + 1\] and \[F(-x) = -2x + 1\].Since \(F(-x) eq F(x)\), the function is not even.

Step 4: Test for Odd Function

Compare \(F(-x)\) with \(-F(x)\):Calculate \(-F(x)\):\[-F(x) = -(2x + 1) = -2x - 1.\]Since \(F(-x) = -2x + 1\) and \(-F(x) = -2x - 1\), \(F(-x) eq -F(x)\), the function is not odd.

Step 5: Determine if the Function is Neither Even nor Odd

Since the function is neither even nor odd based on the above analysis, \(F(x) = 2x + 1\) is neither even nor odd.

Step 6: Sketching the Graph

The function \(F(x) = 2x + 1\) is a linear function with a slope of 2 and a y-intercept of 1. To sketch its graph:- Start at the y-intercept (0,1).- Use the slope to find another point: From (0,1), go up 2 units and to the right 1 unit to reach (1,3).- Draw a line through these points extending in both directions.

Key Concepts

Even and Odd Functions

Understanding the classification of functions as even, odd, or neither can be quite helpful in analyzing their properties and using them effectively in mathematics. Let's break it down simply:

• An even function has symmetry about the y-axis. It satisfies the condition \(f(-x) = f(x)\) for all values of \(x\). This symmetry means that flipping the x-values across the y-axis won't change the function's output.
• An odd function displays symmetry about the origin. It meets the condition \(f(-x) = -f(x)\). If you reflect the graph across both axes, it remains unchanged.
• If a function doesn't follow either condition, it is neither even nor odd. This means there's no simple symmetry present.

Let's take the function from the exercise, \(F(x) = 2x + 1\), as an example. If you substitute \(-x\) for \(x\) and get \(F(-x) = -2x + 1\), you see it doesn't equal \(F(x)\) or \(-F(x)\). That's why \(F(x) = 2x + 1\) is neither even nor odd. This analysis can be done quickly once you practice the substitution and comparison steps.

Linear Functions

Linear functions are some of the simplest and most fundamental functions in algebra. They produce straight lines when graphed and can be represented in the form \(y = mx + b\). Here, \(m\) stands for the slope, and \(b\) is the y-intercept.

• The slope \(m\) tells us how steep the line is and in which direction it tilts. A positive slope means the line goes up from left to right, while a negative slope means it goes down.
• The y-intercept \(b\) is the point where the line crosses the y-axis. It's what y equals when \(x = 0\).

For our function \(F(x) = 2x + 1\), the slope \(m\) is 2. This indicates that for every unit increase in \(x\), \(F(x)\) increases by 2 units. The y-intercept is 1, pinpointing a crossing at the coordinate (0,1). Understanding these parameters helps in quickly visualizing the behavior of a linear function.

Graph Sketching of Functions

Sketching the graph of a function is a crucial skill. It provides a visual representation of how the function operates across different values of \(x\). For linear functions like \(F(x) = 2x + 1\), the process is straightforward:1. **Identify the y-intercept**: Start the sketch at the y-intercept, which is the point on the graph where \(x = 0\). For \(F(x) = 2x + 1\), begin at (0,1).2. **Use the slope**: From the y-intercept, use the slope to find another point on the line. The slope of 2 means go up 2 units for each 1 unit you move to the right. From (0,1), moving to (1,3) reflects this slope.3. **Draw the line**: Connect these points with a straight line extended in both directions. Linear functions are unrestricted like this, continuing infinitely unless specified otherwise.This simple method can assist in graphically representing any linear equation. Practicing graph sketching aids in understanding the structure and properties of different types of functions.