Understanding the Distinctions Between f(x), f(2x), and 2f(x) Functions

To understand the distinctions between the functions f(x), f(2x), and 2f(x), it is essential to break down what each function represents and how they relate to each other. These transformations are fundamental in mathematical analysis and computer science, especially when it comes to optimizing functions for various applications.

Breaking Down the Functions

f(x): This is the original function evaluated at x.

f(2x): This represents the function evaluated at 2x. It means that for each input x, you are doubling the value of x before applying the function f. This typically compresses the graph of f horizontally by a factor of 2.

2f(x): This means that you take the output of the function f(x) and multiply it by 2. This results in vertically stretching the graph of f by a factor of 2.

Summary of Differences

Horizontal Transformation

f(2x) compresses the graph of f horizontally.

Vertical Transformation

2f(x) stretches the graph of f vertically.

Example

For a clearer understanding, let’s consider an example with a specific function such as f(x) x^2:

f(x) x^2 f(2x) 2x^2 4x^2 (compressed horizontally) 2f(x) 2x^2 2x^2 (stretched vertically)

In this example, f(2x) is a horizontal compression by a factor of 2, while 2f(x) is a vertical stretch by a factor of 2.

Function Adjustments and Transformations

Let f:[-11] to R be an arbitrary function that takes as input a real number in the interval [-1, 1] and outputs a real number. Then let’s look at gx f(2x). This is not at all the same function and is only defined for x in [-1/2, 1/2]. As x varies from -1 to 1, f(x) takes on a set of values. Note that g(x) takes on that exact same set of values as x varies from -1/2 to 1/2. That is, if we “squeeze” f(x) so that it fits into the interval [-1/2, 1/2], we get g(x).

What about hx 2f(x)? Again, this is a new function. For every input x, h(x) takes on exactly double the value of f(x). This amounts to “stretching” f(x) in the up/down-direction.

Equation and Slope Analysis

Suppose we have the following equation such that g(x) x^2. Hence, 2g(x) 2x^2. For h(x) f(2x), we have the following:

g(x) x^2 2g(x) 2x^2 h(x) 2x^2 4x^2

The main difference lies in their slopes! In this specific example, 2g(x) 2x^2 and h(x) 4x^2 have different slopes, reflecting the vertical stretching and horizontal compression.