Hi, first post on here. Sorry this seems like a basic question compared to other posts.
As seen in this graph manipulations of ${x}$ take different paths approaching ${x}$. Are the geometric shapes fundamental to the operations? I’m really trying to get a good core understanding.
Edit for clarity:
Before all points meet at (1, 1) the lines approaching (1, 1) are different. If I use the coordinates (0, y to 0, 1), (0, x to 1) and (1, 1) as a frame of reference I can see that each function relates to a different shape within that frame of reference as x is approaching 1.
Context:
I’m looking at patterns in irrational numbers and their geometric operations. Looking at ${x^2}$ for instance; is this curve a rotation or completely linear fundamentally and curved when concatenated with an exponential function?
I’m not a mathematician so it’s very hard to speak the language fluent enough to get my question across effectively.
It is difficult to see exactly what you are asking, but here is my attempt:
The shape of $y=x^2$ is a parabola. Parabolas all have the same general shape: symmetrical, open at one end, a single extremum (max or min value) and extending to infinity in the other direction. The most general parabola can be written $y = ax^2 + bx + c$. It is called a conic section.
The functions you are considering, except for $x^x$ which is a special case, are called power functions. A power function generally is $y = a x^k$. If the power is $k > 1$ then the function rises faster and faster. If the power is $k < 1$ then it rises slower and slower as $x$ increases moving to the right. The larger $k$ is, the more sharply it curves upward.
An irrational exponent is generally defined as the limit of rational exponents that approach it. So $x^1, x^{1.4}, x^{1.41}, x^{1.414},...$ are functions that approach $x^{\sqrt 2}$.