Table of Contents
Understanding Functions
If we wish to understand nature, we must first understand the concept of a function. The functions allow us to describe natural phenomena in a precise and systematic way. In this sense, a function acts as a tool for interpreting and decoding the patterns we observe in the world. Every function consists of one or more independent variables and exactly one dependent variable. To build a clear foundation, we begin with the simplest type of function, one that has a single independent variable and a single dependent variable. Once this basic structure is understood, more complicated functions become much easier to analyze.
Functions can be expressed through words, formulas, and graphs. Among these, graphs and mathematical formulas are often the most effective for visualizing and understanding the behavior of functions.
In this discussion, we designate x as the independent variable and y as the dependent variable. When we plot points \((x,y)\) in the Cartesian coordinate plane, the set of all allowable values of \(x\) is called the Domain, and the corresponding set of resulting values of \(y\) is called the Range. The domain is represented along the x-axis, and the range is represented along the y-axis however, the axes themselves are not the domain or range. A fundamental requirement for any relation to qualify as a function is that each value of the domain must correspond to exactly one value in the range. If a single input produces more than one output, the relation is not a function.
if we graph this equation then it will look like figure 1.
These types of functions are known as piecewise functions, because the function is defined by different formulas over different intervals of the domain. As we observe in the graph, for values of \(x < -2\), the function follows the expression \(1-x\) Since this portion of the function does not include \(x=-2\), we indicate this by placing an open (hollow) dot at that point.
Next, consider the second part of the function, defined for\(-2\leq x<2\). This interval includes -2 but excludes 2. Therefore, the graph shows a closed (solid) dot at \(x=-2\) and an open (hollow) dot at \(x=2\). For the third part of the function, the formula applies for \(x\geq 2\), so the graph begins at x=2 with a solid dot, indicating that the value at x=2 is included.
We can determine the domain and range of each piece using Figure 1 and Equation 1. For the formula \(1-x\) the domain is \(\{x| x < -2 \}\) or in interval notation \((-\infty, -2) \)The corresponding range is \((-\infty, 3)\) . For the second formula, the domain is \([-2,2)\) and the range is \([0,2]\). For the third formula, the domain is \([2,\infty)\), the range is \([3,\infty)\).
Even and Odd Functions
Functions that satisfy the condition \(f(-x) = f(x)\) are called even functions, while those that satisfy \(f(-x) = -f(x)\)are called odd functions. Some functions may be neither even nor odd.The parity of a function (even, odd, or neither) can also be identified directly from its graph. A function is even if the portion of its graph for \(x\geq 0\) is perfectly reflected across the y-axis.A function is odd if the portion of its graph for \(x\geq 0\) can be rotated by \(180 ^\circ\) about the origin to produce the entire graph of \(f(x)\).
To understand this more clearly, consider the basic trigonometric functions \(sin (x)\) and \(cos (x)\). If we examine the graph of \(sin (x)\) as shown in figure 2, we observe that rotating the part of the graph for \(x\geq0\), \(180^\circ\) about the origin reproduces the complete graph. This confirms that \(sin (x)\) is an odd function.
In the graph of \(cos (x)\) shown in Figure 3, the portion of the curve for \(x\geq0\) is reflected symmetrically across the y-axis. This symmetry confirms that \(cos (x)\) is an even function.
Increasing and Decreasing Functions
This concept is extremely useful when analyzing the behavior of a graph. Suppose we select two points \(x_1\) and \(x_2\) and \(x_1< x_2 \) on the x-axis with \(f(x_1) < f(x_2)\) then the graph is increasing over that interval. if \(f(x_1) > f(x_2)\) then the graph is decreasing over that interval. Let us understand this with an example.
If we examine Figure 3 (the plot of \(cos (x)\)), we notice that \(cos (0) = 1\) and \(cos( \frac{\pi}{2})= 0\). here \(cos (0) > cos(\frac{\pi}{2})\), while \(0<\frac{\pi}{2}\) so it is decreasing here. But in \(cos (\frac{-\pi}{2}) < cos (0)\) where \(\frac{-\pi}{2} < 0\) so, this shows that the graph is increasing.
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