Horizontal Stretches and Compressions of Functions

What happen if we consider changes to the inside of a function? As the figure blow shows that when we multiply a function’s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a horizontal stretch; if the constant is greater than 1, we get a horizontal compression of the function.

WHY ?  Let's take a closer look at the graph. Image that if inside the f(x), the x, is running 2 times faster or 2 times larger than the original x, the output f(x) will be 2x larger ?? The answer is NO. However, it will go 2 times faster to arrive the output that originally f(x) will reach. For example, for each output f(x), f(2x) only need 1/2 x to reach the output that f(x) does. In short, For each of the same output f(x), the inside change x, 2x, or 0.5x decide how fast the output f(x) will be reached.

Reference

Abramson, J. (2017). Algebra and trigonometry. OpenStax, TX: Rice University. Retrieved from https://openstax.org/details/books/algebra-and-trigonometry