A function without transformations typically refers to the basic form of a function that is expressed in its standard or simplest format, without any alterations such as shifts, stretches, reflections, or compressions. Here's an example of a basic function:
Linear Function
The simplest form of a linear function is:
\[ f(x) = x \]
This function has a slope of 1 and passes through the origin (0,0). Its graph is a straight line.
Quadratic Function
A basic quadratic function would be:
\[ f(x) = x^2 \]
This function is a parabola that opens upwards with its vertex at the origin.
Trigonometric Function
One of the simplest trigonometric functions is:
\[ f(x) = \sin(x) \]
This function oscillates between -1 and 1 with a period of \(2\pi\).
Exponential Function
A basic exponential function is:
\[ f(x) = e^x \]
This function grows rapidly as \(x\) increases and approaches zero as \(x\) approaches negative infinity.
Summary
Functions in their basic forms serve as the foundation for understanding more complex transformations and variations. Each of these functions can be modified through transformations, but without transformation, their graphs and behaviors retain their fundamental properties. If you have a specific question or need further examples, feel free to ask!
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