Transcendental Functions

These functions cannot be expressed as a finite combination of algebraic operations. They are defined by their unique growth behaviors.

1. Exponential Functions ($e^x, a^x$)

$f(x) = a \cdot b^x$

• Behavior: Starts flat on one side, explodes to infinity on the other.
• Key Feature: The rate of change is proportional to the value itself.
• Common Bases:
  • $e \approx 2.718$ (The natural base)
  • $2$ (Doubling)
  • $10$ (Orders of magnitude)

2. Logarithmic Functions ($\ln x, \log x$)

$f(x) = a \cdot \ln(x) + b$

• Behavior: The inverse of exponential. Grows broadly but extremely slowly.
• Domain: Only defined for $x > 0$ (vertical asymptote at $x=0$).
• Key Point: Usually passes through $(1, 0)$ because $\ln(1) = 0$.

🧠 Advanced Guessing Strategies

Strategy 1: The Growth Hierarchy

When $x$ gets very large ($x \to \infty$), functions "rank" by speed:

1. Exponentials ($e^x$): The fastest. Even $1.01^x$ will eventually beat any polynomial.
2. Polynomials ($x^n$): Fast, but beaten by exponentials.
3. Logarithms ($\ln x$): The slowest. Even $x^{0.1}$ will eventually beat $\ln x$.

Test: If the function shoots up faster than you can scroll, guess Exponential. If it keeps going up but gets flatter and flatter (without strictly leveling off like a horizontal asymptote), guess Logarithm.

Strategy 2: Checking for $e$

The base $e$ is special in Calculus, so it's the default in Function Guessr.

• Check $x = 1$.
  • If $f(1) \approx 2.718$, it's likely just $e^x$.
  • If $f(1) = 2$, it might be $2^x$.

Strategy 3: Domain Constraints

• If the graph disappears for $x < 0$, it is almost certainly a Logarithmic function (or a square root, which is technically algebraic but behaves similarly in domain).
• If the graph exists everywhere but flattens out to $y=0$ on the left side, it's typically Exponential ($e^x$).

Strategy 4: Hyperbolic Functions (Bonus)

Sometimes you might see specific combinations of exponentials:

• $\sinh(x)$: $(e^x - e^{-x})/2$. Looks like cubic $x^3$ but grows much faster.
• $\cosh(x)$: $(e^x + e^{-x})/2$. Looks like quadratic $x^2$ (a hanging chain) but grows faster.