Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The function \( F(x) = 2x + 1 \) is neither even nor odd.

Step 1: Determine Function Symmetry

To check if the function \( F(x) = 2x + 1 \) is even, odd, or neither, we use the definitions: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \), and odd if \( f(-x) = -f(x) \) for all \( x \). First calculate \( F(-x) \).

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

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

Step 3: Check for Even Function Symmetry

Compare \( F(-x) = -2x + 1 \) to \( F(x) = 2x + 1 \). Since \( F(-x) eq F(x) \), the function is not even.

Step 4: Check for Odd Function Symmetry

Compare \( F(-x) = -2x + 1 \) to \(-F(x) = -(2x + 1) = -2x - 1 \). Since \( F(-x) eq -F(x) \), the function is not odd.

Step 5: Conclusion on Symmetry

Since the function is neither even nor odd, classify it as neither. There is no symmetry about the y-axis or the origin.

Step 6: Sketch the Graph

The function \( F(x) = 2x + 1 \) is a linear equation, which can be graphed as a straight line. The y-intercept is 1, and the slope is 2, meaning the line rises 2 units for every unit it moves to the right.

Key Concepts

Function Symmetry

Function symmetry refers to the property of a function where it is either identical on both sides of its graph in relation to an axis, or the origin, or not symmetrical at all. We assess symmetry primarily to understand the nature and behavior of functions better. Symmetry can be one of three types:

• Even Functions: A function is classified as even if it fulfills the condition \( f(-x) = f(x) \) for every value of \( x \). Graphically, this means the function can be folded along the y-axis, and the two halves of the graph will coincide perfectly.
• Odd Functions: A function is odd if \( f(-x) = -f(x) \). In this case, the graph can be rotated 180 degrees about the origin to match onto itself.
• Neither: If a function is neither even nor odd, it lacks specific symmetry. This doesn’t mean it doesn’t have any symmetry; it just doesn't fit the specific criteria for even or odd functions.

To determine the symmetry of a function like \( F(x) = 2x + 1 \), calculate \( F(-x) \) first and compare it with \( F(x) \) and \(-F(x)\). If it doesn't satisfy the criteria for even or odd, like in our example, it's classified as neither.

Linear Functions

Linear functions are one of the most fundamental types of functions, represented by the equation \( y = mx + b \), where "m" is the slope, and "b" is the y-intercept. These functions graph as straight lines, making them easy to predict and analyze.

• Slope (m): The slope determines the steepness and direction of the line. A positive slope means the line increases as it moves from left to right, while a negative slope means it decreases.
• Y-intercept (b): This is where the line crosses the y-axis. It represents the value of \( y \) when \( x = 0 \).

In the example \( F(x) = 2x + 1 \), the slope is 2, indicating the line is fairly steep, rising two units for each unit moving right. The y-intercept is 1, which tells us that the line crosses the y-axis at the point (0, 1). Understanding slope and y-intercept helps in quickly sketching the graph of any linear function.

Graphing Functions

Graphing is a visual way to understand and interpret mathematical functions. When graphing linear functions, the process involves identifying key components like the slope and y-intercept to plot the function accurately.

• Identifying Key Points: Start by plotting the y-intercept on the coordinate plane, which gives the initial point of the function on the y-axis.
• Applying the Slope: From the y-intercept, use the slope to find another point. If the slope is \( \frac{rise}{run} \), move up/down and left/right accordingly to plot the next point.
• Drawing the Line: Once two points are established from the slope and y-intercept, draw a straight line through them to represent the function.

For \( F(x) = 2x + 1 \), begin by placing a point at (0, 1) on the graph. Then, since the slope is 2, move up 2 units and right 1 unit to place the next point. Connect these dots with a straight line, and extend it across the graph. This is a straightforward way to visualize the function and understand its behavior.