Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(g(x)=x\)

Step-by-Step Solution

Answer

The graph of the function \(g(x) = x\) is a straight line passing through the origin and the function is odd.

Step 1 - Sketching the Graph

The graph of \(g(x) = x\) is a straight line passing through the origin with a slope of 1. It increases as x increases and decreases as x decreases. This means it goes from the lower left to the upper right.

Step 2 - Check for Even Function

An even function is one in which \(g(x) = g(-x)\) for all x. In the case of the function \(g(x) = x\), \(g(-x) = -x\), which is not equal to \(g(x)\) so the function is not even.

Step 3 - Check for Odd Function

An odd function is one in which \(g(-x) = -g(x)\) for all x. In the case of the function \(g(x) = x\), \(g(-x) = -x\) which is equal to \(-g(x)\), thus the function is odd.

Key Concepts

Even and Odd Functions

Understanding whether a function is even, odd, or neither is a fundamental aspect of analyzing its symmetry. An even function is symmetrical around the y-axis, like reflecting an image in a mirror. Mathematically, a function f is even if for every number x, the equality f(x) = f(-x) holds true. For example, the function f(x) = x^2 is even because squaring a number yields the same result whether the number is positive or negative.

In contrast, an odd function has rotational symmetry around the origin, meaning if the graph were to be rotated 180 degrees about the origin, it would look the same. A function f is odd if the equation f(-x) = -f(x) is valid for all x. A classic example of an odd function is f(x) = x^3. In the exercise, the function g(x) = x satisfies the condition for an odd function since g(-x) = -g(x) is upheld. Therefore, students can determine that the function has rotational symmetry and is odd.

Remembering this distinction can significantly simplify the graphing process and enhance understanding of the function's behavior across the entire coordinate plane.

Graph Sketching

Sketching the graph of a function is a visual way to represent its behavior and is crucial for understanding its characteristics. When graphing a linear function such as g(x) = x, the process begins by identifying key features such as the slope and y-intercept. For g(x) = x, the slope is 1, and the y-intercept is at the origin (0,0). This means for every step to the right (increase in x), the value of g(x) increases by the same amount. The graph is a straight line that bisects the first and third quadrants, rising diagonally from the bottom left to the top right of the graph.

When teaching graph sketching, it's helpful to focus on plotting points and using the slope, or the 'rise over run' technique, to ensure accuracy. For functions that are more complex, identifying symmetry, intercepts, and asymptotes is also important. Creating a table of values can aid in plotting points for less straightforward functions, and always checking the function's behavior as x approaches positive and negative infinity gives insight into the graph's end behavior.

Linear Function Properties

Linear functions are the most basic and fundamental type of function in algebra, often represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The properties of linear functions are straightforward, making them excellent for foundational understanding.

One of the key characteristics of a linear function is that it has a constant rate of change, which means the slope is the same between any two points on the line. In the function g(x) = x, the slope (or rate of change) is 1. This uniformity results in a graph that is a straight line. Another important property is that linear functions have exactly one y-value for every x-value, which makes them functions—this is known as the vertical line test.

Moreover, linear functions are neither bounded above nor below, which implies that the outputs f(x) can be infinitely large or small. In practical terms, understanding the properties of linear functions equips students to analyze and predict trends and relationships in various real-world situations, from simple budgeting to complex engineering problems.