Problem 7 - Precalculus with Limits

Question

A function \(f\) is ________ if, for each \(x\) in the domain of \(f, f(-x) = -f(x)\).

Step-by-Step Solution

Answer

The function is Odd

Step 1: Understanding the exercise

The task is to complete the blank with the type of function where \(f(-x) = -f(x)\) for every \(x\) in the function’s domain. This equation explains a kind of symmetry a function can possess.

Step 2: Identifying the property of the function

Analyze the given statement \(f(-x) = -f(x)\). This indicates that the function is symmetric with respect to the origin. That’s because, if you replace \(x\) with \(-x\), the sign of the whole function flips, replicating a rotation of 180 degrees around the origin.

Step 3: Conclusion

A function satisfying the equation \(f(-x) = -f(x)\) is referred to as an odd function. So, the blank in the exercise will be filled with the term 'odd'.

Key Concepts

Function Symmetry

Function symmetry is a fascinating concept in mathematics that helps us understand how a function behaves concerning specific points or axes. Essentially, it gives us a way to tell if the function’s graph looks the same after flipping or rotating. There are two main types of symmetry associated with functions:

• Symmetry about the y-axis: If for every point \((x, y)\) on the graph of a function, there's a matching point \((-x, y)\), the function is symmetric about the y-axis. This kind of symmetry is typical for even functions.
• Symmetry about the origin: If for each \((x, y)\) on a function’s graph, there's a matching point \((-x, -y)\), the function is symmetric about the origin. Odd functions exhibit this type of symmetry.

Symmetry can be visually appealing and makes calculating function values easier, as symmetry simplifies the domains that need to be individually considered.

Properties of Functions

Understanding properties of functions is crucial, as they help us define how the function behaves under different operations or transformations. When we talk about properties, we may consider:

• Domain and Range: The domain is each value that can go into a function. The range is the set of possible output values.
• Continuity: A function is continuous if there are no breaks, jumps, or holes in its graph.
• Growth behavior: This refers to how a function increases or decreases, which can be constant, linear, exponential, etc.
• Symmetry: As discussed, even functions show y-axis symmetry, while odd functions have origin symmetry.

These properties aid in sketching the function and predicting its behavior across its domain, providing insights before calculating specific values.

Even and Odd Functions

Even and odd functions are fundamental classifications that tell us how a function responds when its input sign is reversed. They help in understanding the algebraic and geometric behavior of functions:

• Even Functions: A function \(f\) is even if \(f(-x) = f(x)\) for all \(x\) in its domain. Geometrically, this means its graph is symmetric about the y-axis, as the values mirror each other about this axis. Common examples are \(x^2\) and \(\cos(x)\).
• Odd Functions: A function \(f\) is odd if \(f(-x) = -f(x)\) for all \(x\) in its domain. This implies the function is symmetric about the origin—think of turning the graph 180 degrees around the origin point. Examples include \(x^3\) and \(\sin(x)\).

These distinct types of functions simplify analysis and often appear in solutions to problems related to equations, integrations, and other mathematical operations.