Problem 76 - Algebra and Trigonometry

Question

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=x^{3} $$

Step-by-Step Solution

Answer

The function \(f(x) = x^3\) is odd.

Step 1: Understand the definitions

A function is **even** if for all x in the domain, \( f(-x) = f(x) \). A function is **odd** if for all x in the domain, \( f(-x) = -f(x) \). If neither of these hold, the function is neither even nor odd.

Step 2: Substitute and check for evenness

Substitute \(-x\) into the function \( f(x) = x^3 \) and simplify. \[ f(-x) = (-x)^3 = -x^3 \].Check if \( f(-x) = f(x) \). Since \(-x^3 eq x^3\), the function is not even.

Step 3: Substitute and check for oddness

Compare \( f(-x) \) to \(-f(x)\).\[ -f(x) = -x^3 \]. Since \( f(-x) = -f(x) \), the function is odd.

Step 4: Conclude and use symmetry to sketch

The function \( f(x) = x^3 \) is odd, meaning its graph is symmetric about the origin. Sketch the graph by plotting points such as (-2,-8), (-1,-1), (0,0), (1,1), and (2,8), and draw a smooth curve through these points maintaining origin symmetry.

Key Concepts

Symmetry in Functions

Symmetry plays an important role in understanding the nature of functions. A function can be classified based on its symmetry properties.
When a function shows symmetry in its graph, it offers a visual balance which can simplify the understanding of its behavior and characteristics. There are mainly two types of symmetries related to functions: symmetry about the y-axis and symmetry about the origin.

• **Symmetry about the y-axis** indicates that the function is even. This means that if you flip the function graph across the y-axis, it would remain unchanged.
• **Symmetry about the origin** is characteristic of odd functions. An odd function, when you rotate its graph by 180 degrees around the origin, appears unchanged.

Understanding these symmetries helps when sketching graphs, allowing you to predict parts of the graph based on its symmetry properties. When a graph has no symmetry, it's neither even nor odd, implying no defined reflection or rotation properties.

Even Functions

Even functions have a special property: for every x value in the function's domain, the value at \( -x \) will be the same as at \( x \). This is written mathematically as \( f(-x) = f(x) \).
One can easily identify even functions on a graph by looking for symmetry about the y-axis. Examples of even functions are \( f(x) = x^2 \) and \( f(x) = \cos(x) \). When graphed, these functions produce a mirror image across the y-axis.
Even functions often arise in situations where the input's positive and negative values yield the same results, which is common in certain types of physical and mathematical phenomena. Knowing how to identify even functions helps in analyzing real-world scenarios where symmetry plays a role.

Graphing Function Behavior

Function graphs provide a powerful visual representation of mathematical concepts. They help in understanding how different types of functions behave.

• For odd functions, like \( f(x) = x^3 \), the graph is symmetric about the origin. This means if you rotate the graph 180 degrees around the origin point of the graph, it looks the same.
• To sketch a graph of \( f(x) = x^3 \), you can plot points such as (-2,-8), (-1,-1), (0,0), (1,1), and (2,8). These points show how the function affects both negative and positive values of x. As you connect these points with a smooth curve, you'll notice it's symmetric around the origin, embodying the characteristics of odd functions.

Graphing is a crucial skill in mathematics as it allows students to visualize mathematical principles and makes complex ideas easier to grasp.