Поширені функції: що та чому?

We've been studying polynomials, exponential functions, logarithms, and other functions for a long time: there was evidence that even the ancient Egyptians were solving equations like $ax+b=0$ or $ax^2+bx=c$. But why do we study functions like $\color{blue}{f(x)=ax+b}$, $\color{green}{g(x)=ax^2+bx+c}$, or $\color{red}{h(x)=ab^x}$?

The simple answer? Because we felt like it.

Polynomials are just a series of repeated multiplication, addition, and subtraction. They are just so exceedingly simple, it's extremely hard to go through more than 3000 years worth of mathematics without even just accidentally bumping into them.

Now, an ancient mathematician might suddenly wonder: If I had $12$ sheep, and bought $8$ more, then I would have twice the amount of sheep than my neighbor, how many sheep does my neighbor have? The equation looks like this: $2x=12+8$ or $$2x-8=12$$ (to emphasize my point). Bam, we have a polynomial!

Now suppose we had a plot of land $10$ feet by $10$ feet (or whatever other unit said ancients use) and we annexed $2$ feet on both sides. What is the size of the new land? If we had $x$ feet by $x$ feet, then the size of land would be $(x+2)^2=x^2+4x+4$. Another polynomial!

Soon enough, when Algebraic notation was invented and polynomials studies more extensively, we found out that our medieval ancestors' intuition about polynomials was spot-on. Polynomials found a home in many real-world situations, along with endlessly assisting mathematicians in their further endeavors. For example, functions were found to be able to be approximated by polynomials (and represented by power series): $$\sin x=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}+\cdots$$

This way, we could easily find the value of, say, $\sin 2.4$ by just plugging it into the easy-to-compute polynomial approximation!

What about exponential functions? Why would we ever study something with a variable in the exponent?

Many of you should already know one (of many) answer if you've ever invested some money into a local bank. Interest!

When money is invested into the bank, the interest increases the amount of money exponentially, if only very-slow-exponentially. For example, if you put $\$1000$ into your local bank, which has an interest of $5\%$ per year (wow!), then the amount of money you have in $t$ years can be expressed by the equation $$f(t)=1000(1.05)^t$$

In addition, exponential functions can be used to express the population of microorganisms like bacteria. If a sample has $1$ bacterium at time $t=0$, and each bacterium splits into two bacteria every second, then the population of bacteria expressed as a function of time is $$f(t)=2^t$$

But how about logarithm functions? Surely, they must be much more useless than their exponential counterparts.

Logarithms were invented around the 1500's for making multiplication, division, and square roots easier. If you had to multiply, say, $1912$ and $4937$ together, it would take a lot of time and tedious work. However, taking the logarithms of both numbers using a logarithm table, adding these two logarithms, and then converting back by taking the exponent turned the grueling multiplication into the much easier addition! The time it took to calculate with large numbers dramatically reduced.

In the academics, logarithms are also extremely popular. For example, the distribution of primes in the positive integers can be approximated by the function $$f(x)=\dfrac{x}{\ln x}$$ where we are checking for primes in the range $1\to x$. Also, logarithms are used to find the pH of a solution, given the hydrogen ion activity ($a_{H^+}$): $$pH=-\log a_{H^+}$$

As you have seen, the special functions we learn aren't just arbitrary; they serve to give us a deeper understanding of mathematics, along with the world around us. Be assured, the things you learn in Middle School and High School will not be in vain (entirely)!