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Magazine Chapter 0: Problem 17
Specify whether the given function is even, odd, or neither, and then sketch its graph. $$F(x)=2 x+1$$
Short Answer
Expert verified
The function is neither even nor odd.
Step by step solution
01
Understand the Definitions
To determine if a function is even or odd, we use the definitions: A function \(f(x)\) is even if \(f(-x) = f(x)\) for all \(x\) in the domain, and a function is odd if \(f(-x) = -f(x)\). If neither condition is met, the function is neither even nor odd.
02
Calculate \(F(-x)\)
Substitute \(-x\) into the function to find \(F(-x)\). For \(F(x)=2x+1\), we have \(F(-x)=2(-x)+1=-2x+1\).
03
Compare \(F(-x)\) with \(F(x)\)
Compare \(F(-x) = -2x + 1\) with \(F(x) = 2x + 1\). Since \(-2x + 1 eq 2x + 1\) and \(-2x + 1 eq -(2x + 1)\), the function does not satisfy the conditions for being even or odd.
04
Conclude Whether the Function is Even, Odd, or Neither
Since \(F(-x)\) is neither equal to \(F(x)\) nor equal to \(-F(x)\), the function \(F(x) = 2x + 1\) is neither even nor odd.
05
Sketch the Graph of \(F(x) = 2x + 1\)
To sketch the graph, note that \(F(x) = 2x + 1\) is a linear function with a slope of 2 and a y-intercept at (0,1). Draw a straight line passing through the point (0,1) with an upward slope of 2. It will cross the x-axis at \((-0.5, 0)\). This is a straight line graph with no symmetry about the y-axis or origin, consistent with the function being neither even nor odd.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Functions
Linear functions are a fundamental building block in mathematics. They are expressed in the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Linear functions graph as straight lines, so their behavior is predictable and consistent.
- The Slope: The slope \(m\) determines the steepness of the line, indicating how much \(y\) increases or decreases as \(x\) increases by 1 unit. A positive slope results in an upward line, while a negative slope results in a downward line.
- The Y-Intercept: This is the point where the line crosses the y-axis, represented by \(b\). It indicates the value of \(f(x)\) when \(x = 0\).
Even and Odd Functions
In calculus and algebra, functions can often be classified as even, odd, or neither. This classification helps understand the symmetry of the graph.
- Even Functions: A function \(f(x)\) is even if \(f(-x) = f(x)\). This symmetry means the graph reflects identically across the y-axis, like \(x^2\).
- Odd Functions: A function is considered odd if \(f(-x) = -f(x)\), indicating rotational symmetry about the origin \((0,0)\), like \(x^3\).
Graphing Functions
Graphing functions is a crucial skill that allows us to visualize the behavior and characteristics of mathematical equations. For linear functions, the process typically involves a few key steps.
- Identify Key Components: Determine the slope \(m\) and y-intercept \(b\) from the linear function.
- Plot the Y-Intercept: Start graphing by marking the y-intercept on the y-axis.
- Use the Slope: Using the slope \(m\), move from the y-intercept. If the slope is positive, move up and to the right; if negative, move down and to the right.
- Draw the Line: Extend the line through the points plotted, ensuring it reflects the linear nature of the function.
