Problem 62 - Algebra and Trigonometry

Question

61–68 ? 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 Problem

We are tasked with determining if the function \( f(x) = x^{-3} \) is even, odd, or neither. A function \( f \) is considered even if \( f(-x) = f(x) \) for all \( x \) in the domain. It is considered odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.

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

To determine whether the function is even or odd, we need to calculate \( f(-x) \). Substitute \(-x\) into the function: \( f(-x) = (-x)^{-3} \). Simplifying this expression gives us \( f(-x) = -1 \cdot x^{-3} = -x^{-3} \).

Step 3: Compare \( f(-x) \) with \( f(x) \)

Now that we have \( f(-x) = -x^{-3} \), compare it with \( f(x) = x^{-3} \). We observe that \( f(-x) = -f(x) \), which means the function satisfies the condition for being odd.

Step 4: Conclusion about Function

Since \( f(-x) = -f(x) \), the function \( f(x) = x^{-3} \) is odd. Odd functions have rotational symmetry about the origin.

Step 5: Sketch the Graph (Optional)

For an odd function like \( f(x) = x^{-3} \), the graph is symmetric with respect to the origin. This means if you rotate the graph 180 degrees about the origin, it would look the same. The graph of \( f(x) = x^{-3} \) is similar to a hyperbola, with branches in the first and third quadrants, approaching but never touching the axes.

Key Concepts

Function Symmetry

Identifying the symmetry in functions helps understand their behavior and sketch their graphs more efficiently. A function can be symmetric in different ways: even, odd, or none at all. Here's a quick overview:

• **Even Functions**: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain. These functions are symmetric about the y-axis. Think of the classic parabola \( f(x) = x^2 \) as a good example.
• **Odd Functions**: A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \). They have rotational symmetry about the origin, meaning if you rotate their graph 180 degrees, it remains unchanged. A perfect example is \( f(x) = x^3 \).
• **Neither**: Some functions don't fall into either category, like \( f(x) = x + 1 \). They are neither even nor odd. This means no specific symmetry patterns can be found.

Understanding these symmetries provides a powerful tool in graph sketching. They reveal how parts of a graph mirror or rotate without the need for plotting every point.

Function Transformation

Function transformations modify the appearance of graphs, often making them shifted, stretched, or reflected. Knowing these transformations can simplify the graph sketching process.

• **Vertical Shifts**: Changing values like \( f(x) + c \) moves the graph up if \( c > 0 \) and down if \( c < 0 \).
• **Horizontal Shifts**: Modifications such as \( f(x + c) \) move the graph left if \( c > 0 \) and right if \( c < 0 \).
• **Reflections**: Transformations like \( -f(x) \) reflect the graph over the x-axis, while \( f(-x) \) reflects over the y-axis.
• **Stretches and Compressions**: Multiplying the function as \( c \, f(x) \) stretches it vertically if \( c > 1 \) and compresses if \( 0 < c < 1 \). Horizontally, values within the argument like \( f(c \, x) \) have the opposite effect.

By combining transformations, you can manipulate basic graphs to model complex behaviors. This adaptability is crucial in graph sketching and understanding real-world scenarios.

Graph Sketching

Graph sketching is the skill of drawing accurate plots of functions based on properties, transformations, and symmetries without plotting numerous points. Here's how you can approach it effectively:

• **Determine Symmetry**: Check if the function is even, odd, or neither. For \( f(x) = x^{-3} \), since it's odd, recognize its origin symmetry first.
• **Assess Behavior**: Identify intercepts, extreme points, or asymptotes. \( f(x) = x^{-3} \) behaves similarly to a hyperbola, with its tails approaching but never touching the axes.
• **Apply Transformations**: Consider any shifts or stretches. For \( x^{-3} \), note how modifications like \( -x^{-3} \) influence the graph.
• **Sketch**: Start with symmetry identified. For odd functions like our example, sketch one quadrant then use rotational symmetry to complete.

Graph sketching goes beyond memorizing shapes. It entails understanding functions so deeply that visualizing how they'll look becomes intuitive. Practice with known functions before tackling complex equations.