>[!abstract] >In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples include $f(x) = 1/x$ and $f(x) = \sqrt(x)$. The purpose of this note is to list the differences between polynomial, exponential, logarithmic, and rational functions. ## Polynomial functions The variable ($x$) is in the base. The general form is $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$ where $a_n$, $a_{n-1}$, ..., $a_0$ are *coefficients*, $n$ is a *non-negative* integer (and the highest exponent determines the *degree* of the polynomial), and $x$ is the variable. | Name | Value of $n$ (degree) | Form | | --------- | --------------------- | ------------------------------------------- | | Constant | $n = 0$ | $f(x) = a_0$ | | Linear | $n = 1$ | $f(x) = ax + b$ | | Quadratic | $n = 2$ | $f(x) = ax^2 + bx + c$ | | Cubic | $n = 3$ | $f(x) = ax^3 + bx^2 + cx + d$ | | Quartic | $n = 4$ | $f(x) = ax^4 + bx^3 + cx^2 + dx + e$ | | Quintic | $n = 5$ | $f(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$ | | ... | ... | ... | | General | $n \ge 0$ | $f(x) = a_nx^n + \ldots + a_0$ | ## Exponential functions The variable ($x$) is in the exponent. The general form is $f(x) = a \cdot b^x$ where $a$ is the initial value (or *vertical stretch factor*), $b$ is a positive constant ($b \gt 0$ and $b \neq 1$), and $x$ is the variable exponent. - If $b \gt 1$, the function increases as $x$ increases (*exponential growth*); for example, $f(x) = 2^x$. - If $0 \lt b \lt 1$, the function decreases as $x$ increases (*exponential decay*); for example, $f(x) = (1/2)^x$. ## Logarithmic functions They are the inverse of exponential functions. The general form is $f(x) = a \log_b(x) + c$ where $a$ and $c$ are constants, $b$ is the base of the logarithm ($b \gt 0$ and $b \neq 1$), and $x$ is the input (must be positive, because logarithms are undefined for $x \le 0$). So if $y = b^x$ is an exponential function, the logarithmic function $x = \log_b(y)$ reverses it. For example, $f(x) = 10^x$ is the exponential function and $g(x) = \log_{10}(x)$ is its logarithmic function. ## Rational functions The exponent is either fractional or negative (which can be written as a reciprocal fraction). The general form if fractional: $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are each a polynomial function, and always non-negative or always non-positive at the same time (so that their quotient is positive). The general form if negative is: $f(x) = ax^{-n} = \frac{a}{x^n}$. >[!related] >- **North** (upstream): — >- **West** (similar): — >- **East** (different): — >- **South** (downstream): —