Table of Contents
Introduction
Functions form the backbone of modern mathematics and the sciences. From the earliest developments in scientific thought, functions have served as the essential tools that allow us to describe, analyze, and predict real world phenomena.
They convert intuition into structure, and structure into clear, reliable results. In the simplest sense, a function can be viewed as a mathematical rule or formula that associates every valid input with a specific output.
Yet, in practice, functions are far more than rules they are the language through which we understand relationships, change, motion, growth, and countless other aspects of the natural world.
In this blog, we will explore how functions are constructed and interpreted using both graphical and mathematical methods. These approaches together provide a powerful foundation for understanding real world systems and for building accurate models.
We will study linear functions in detail, examining how linear function arises naturally in scientific and everyday contexts. More advanced categories power functions, trigonometric functions,polynomial functions, exponential functions, and logarithmic functions will be treated thoroughly in the next blogs.
Our goal is not simply to define functions, but to develop the intuition, visual understanding, and analytical skills needed to use them confidently and effectively.If you want to understand functions basic you can read this blog.
Linear function
A linear function describes a relationship between two variables: one independent variable and one dependent variable. The behavior of this relationship is controlled by two constants, which determine how the graph of the function is positioned and oriented in the Cartesian coordinate system.
When the independent variable is multiplied by a constant, the resulting change affects the vertical variation of the line. This constant is called the slope, the slope measures how steep the line is.
Another constant is added to the expression to shift the entire line horizontally within the coordinate plane. This constant is known as the intercept, more precisely the y-intercept, because it indicates where the graph crosses the y-axis.
Mathematically, every linear function can be written in the form \[y = mx + c\] where y and x are dependent and independent variables respectively. m and c are slope and intercept respectively. To understand this more clearly, consider the linear function \( y= 8x +3\) If we take a few values of x and compute the corresponding values of y (as shown in Table 1), then plot the ordered pairs on a coordinate plane, we obtain the straight line shown in Figure 1.
From the Table 1 and Figure 1, you will notice that for every 1 unit increase in x, the value of y increases by 8 units. This constant rate of change 8 in this case is the slope. Thus, the slope can be interpreted as the rate of change of y with respect to x. Here we can also use this function to predict the value of y at any value of x. suppose we want to find the value of y at x = 8, y must be 67 .
As of now we learn how a function can be used in real life to interpret and predict. we will learn advance versions of linear functions and thier use in machine learning.
Problem
Pressure increases at a constant rate with depth, the relationship is linear. We can use the slope formula \[m= \frac{y_2 -y_1}{x_2 – x_1}\] here we have given two points \((0,14.7)\) and \((10, (14.7+4.45))\) which is \((10,19.15)\). so slope is \[m= \frac{19.15-14.7}{10-0}\] \[m= 0.445\] according to point slope equation \[(y-y_1)=m(x-x_1)\] we can write \[(y-14.7)=0.445(x-0)\] \[y=0.445x+14.7\].
From this equation we can interprit that when ever depth\((x)\) is 0, the pressure \((y)\) is \(14.7\frac{lb}{in^2}\) and slope indicate that for every feet change in depth there is 0.445 unit change in pressure.
we will predict the pressure at a depth of \(1000ft\) \[y= (0.445\times 1000)+14.7\] \[y= 459.7 \frac{lb}{in^{2}}\]
Now we want, what will be the depth when pressure reaches \(100 \frac{lb}{in^{2}}\).
\[x=\frac{100-14.7}{0.445}\] \[x= 191.6853 ft\]
so at 191.6853ft the pressure will be \(100 \frac{lb}{in^{2}}\).