Pre-Calculus Review

This is the capstone review for Pre-Calculus. It brings together everything you’ve covered, from advanced functions through limits, with key formulas, mixed examples, and connections to what comes next in Calculus.

Function composition takes the output of one function and feeds it into another. If f(x) = 2x + 1 and g(x) = x², then:

$$(f \circ g)(x) = f(g(x)) = 2x^2 + 1$$ $$(g \circ f)(x) = g(f(x)) = (2x + 1)^2 = 4x^2 + 4x + 1$$

Order matters. Composition is not commutative.

Inverse functions “undo” each other. If f(x) = 3x - 2, then its inverse swaps x and y and solves:

$$f^{-1}(x) = \frac{x + 2}{3}$$

Piecewise functions use different rules on different intervals. Absolute value functions are a special case where the function “bounces” at a point.

End behavior depends on the degree and leading coefficient. Even degree with positive leading coefficient: both ends go up. Odd degree with positive leading coefficient: left goes down, right goes up.

The Fundamental Theorem of Algebra says a degree-n polynomial has exactly n zeros (counting multiplicity and complex zeros). Complex zeros of polynomials with real coefficients always come in conjugate pairs: if a + bi is a zero, so is a - bi.

Rational functions can have vertical asymptotes (where the denominator is zero and doesn’t cancel), horizontal asymptotes (determined by comparing degrees), and holes (where a factor cancels).

The key relationship:

$$y = b^x \iff x = \log_b y$$

The natural logarithm ln(x) uses base e (approximately 2.718). Continuous growth and decay use the formula:

$$A = Pe^{rt}$$

The double-angle identities extend the sum formulas:

$$\sin 2\theta = 2\sin\theta\cos\theta$$ $$\cos 2\theta = \cos^2\theta - \sin^2\theta$$

Polar coordinates represent points using distance and angle (r, θ) instead of (x, y). Conversion formulas:

$$x = r\cos\theta, \quad y = r\sin\theta, \quad r = \sqrt{x^2 + y^2}$$

Polar curves include cardioids, roses, limacons, lemniscates, and spirals.

Vectors have magnitude and direction. The dot product returns a scalar:

$$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 = \lvert\mathbf{u}\rvert\lvert\mathbf{v}\rvert\cos\theta$$

Two vectors are perpendicular when their dot product is zero.

Matrix multiplication: row times column. The 2x2 inverse formula:

$$A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

If the determinant (ad - bc) is zero, the matrix has no inverse.

Row reduction (Gaussian elimination) solves systems by converting to row echelon form. Cramer’s Rule uses determinants to find solutions by swapping columns.

Systems can have one solution (lines cross), no solution (parallel lines), or infinitely many solutions (same line).

Arithmetic series sum (constant difference d):

$$S_n = \frac{n}{2}(a_1 + a_n)$$

Geometric series sum (constant ratio r):

$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$$

Infinite geometric series (converges when the absolute value of r is less than 1):

$$S_\infty = \frac{a_1}{1 - r}$$

Mathematical induction proves statements for all positive integers using two steps: prove the base case (n = 1), then show that if it works for k, it works for k + 1.

A limit describes what a function approaches as x gets close to a value:

$$\lim_{x \to a} f(x) = L$$

When direct substitution gives 0/0, simplify first (usually by factoring).

A function is continuous at x = a when three conditions hold: f(a) is defined, the limit exists, and the limit equals f(a). Discontinuities come in three types: removable (holes), jump, and infinite (vertical asymptotes).

The limit laws let you break complicated limits into simpler pieces by adding, subtracting, multiplying, dividing, and raising to powers.

Here are the formulas worth having at your fingertips:

$$\text{Dot product: } \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2$$ $$\text{Geometric sum: } S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$$ $$\text{Infinite geometric: } S_\infty = \frac{a_1}{1 - r}$$ $$\text{Continuous growth: } A = Pe^{rt}$$ $$\text{Double angle: } \sin 2\theta = 2\sin\theta\cos\theta$$

1. Function composition

If f(x) = 2x + 1 and g(x) = x², find (f composed with g)(3):

$$g(3) = 9, \quad f(9) = 2(9) + 1 = 19$$

2. Rational function asymptotes

Find all asymptotes of f(x) = (2x² - 3) / (x² - 4):

Vertical asymptotes where the denominator is zero: x = 2 and x = -2.

Horizontal asymptote: degrees are equal, so y = 2/1 = 2.

3. Infinite geometric series

Sum of 6 + 2 + 2/3 + 2/9 + …

Here a₁ = 6 and r = 1/3. Since the absolute value of r is less than 1, it converges:

$$S_\infty = \frac{6}{1 - \frac{1}{3}} = \frac{6}{\frac{2}{3}} = 9$$

4. Limit with factoring

$$\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$$

Direct substitution gives 0/0. Factor the numerator:

$$= \lim_{x \to 4} \frac{(x-4)(x+4)}{x-4} = \lim_{x \to 4} (x + 4) = 8$$

Everything in Pre-Calculus builds the tools you need for Calculus. Limits become derivatives (instantaneous rates of change). Series become integrals (accumulation). Vectors extend to multivariable calculus. Matrices power linear algebra. The concepts you’ve learned here aren’t just prerequisites. They’re the language that Calculus is written in.