1. What is Function Dilation?

Function dilation is a transformation that stretches or compresses a function horizontally by a factor of k. The formula \( f_d(x) = f\left(\frac{x}{k}\right) \) represents a horizontal dilation of the original function f(x).

2. How Does the Calculator Work?

The calculator uses the dilation formula:

Where:

• \( f_d(x) \) — Dilated function
• \( f(x) \) — Original function
• \( k \) — Dilation factor (dimensionless)

Explanation: When k > 1, the function is stretched horizontally. When 0 < k < 1, the function is compressed horizontally. When k < 0, the function is also reflected across the y-axis.

3. Importance of Function Dilation

Details: Function dilation is fundamental in various mathematical applications including signal processing, image scaling, and geometric transformations. Understanding dilation helps in analyzing how functions behave under scaling operations.

4. Using the Calculator

Tips: Enter the original function f(x) and the dilation factor k. The dilation factor must be a non-zero value. Common functions include polynomials, trigonometric functions, exponential functions, etc.

5. Frequently Asked Questions (FAQ)

Q1: What happens when k = 1?
A: When k = 1, the function remains unchanged as \( f_d(x) = f(x) \).

Q2: How does negative k affect the function?
A: A negative k value causes both dilation and reflection across the y-axis.

Q3: Can I use this for any type of function?
A: Yes, the dilation formula applies to all functions, though the visual effect may vary depending on the function type.

Q4: What's the difference between horizontal and vertical dilation?
A: Horizontal dilation affects the x-values (\( f(x/k) \)), while vertical dilation affects the y-values (\( k \cdot f(x) \)).

Q5: How is this used in real-world applications?
A: Function dilation is used in image processing for scaling, in audio processing for time-stretching, and in physics for scaling transformations.