Functions: Composition, Combination & Algebraic vs Transcendental

Algebraic vs Transcendental Functions

• Algebraic functions are built from a finite sequence of polynomial operations — addition, subtraction, multiplication, division, and roots. Polynomials (like a quadratic $ax^2+bx+c$) and rational functions are algebraic.
• Transcendental functions are everything else: logarithmic, exponential, trigonometric, and hyperbolic functions all count as transcendental.

Worked example: of exponential, hyperbolic, and quadratic functions — the quadratic is the algebraic one (a polynomial); the other two are transcendental.

Composing Functions: $(f\circ g)(x) = f(g(x))$

Always evaluate the inner function first, then feed that result into the outer function.

Worked example: $f(x)=3x+2$, $g(x)=2x+5$, find $(f\circ g)(5)$: g(5) = 2(5)+5 = 15, then f(15) = 3(15)+2 = 47.

Worked example (composing with a formula, not a number): $g(x)=2x+3$, $h(x)=x^2-4$, find h(g(x)): $h(g(x)) = (2x+3)^2-4 = 4x^2+12x+9-4 = 4x^2+12x+5$.

Worked example (a square-root composition): $f(x)=x-2$, $g(x)=\sqrt{x^2+1}$, find $(g\circ f)(x) = g(f(x)) = \sqrt{(x-2)^2+1} = \sqrt{x^2-4x+4+1} = \sqrt{x^2-4x+5}$.

Combining Functions Directly

Functions can also be added, subtracted, multiplied, or divided pointwise — just combine their formulas term by term.

Worked example: $f(x)=x^2-8$, $g(x)=-2-x^2$, find f(x)+g(x): $(x^2-8)+(-2-x^2) = -10$ — a constant function (the x² terms cancel exactly).

The Graph of a Function

The graph of f is the plot of all ordered pairs (x, f(x)) in the coordinate plane — this is the formal term for what you'd otherwise just call "the curve."

Even and Odd Functions

• Even: $f(-x) = f(x)$ for every x in the domain — symmetric about the y-axis.
• Odd: $f(-x) = -f(x)$.

Why This Matters for Your Exams

For any composition question, write out g(the input) first, get a number or expression, and only then substitute that whole result into f — never substitute into both functions simultaneously.