Math 3C Summative Cheat Sheet

This cheat sheet summarizes key concepts from Math 3C, covering inverse functions, logarithmic and exponential functions, and polynomial operations. It provides formulas, properties, and methods for solving equations and analyzing graphs.

Core Principles

Inverse functions satisfy f(g(x)) = x and g(f(x)) = x.
Logarithmic and exponential functions are inverse pairs.
Log properties (product, quotient, power) simplify expressions.
Polynomials can be factored using various algebraic techniques.
Zeros of a polynomial correspond to x-intercepts and dictate graph behavior.

Action Steps

Find inverse: Swap x and y, then solve for y.
Evaluate inverse: If f(a) = b, then f⁻¹(b) = a.
Solve log equations: Condense using properties, convert to exponential form or set arguments equal.
Solve exponential equations: Use same-base exponents or take ln/log of both sides.
Factor polynomials: Always check for GCF first.
Write equation from graph: Use zeros for factors, determine multiplicity from graph behavior (cross/bounce), use a point to find leading coefficient 'a'.

Key Terms

Inverse Function: A function that 'undoes' another function. If f(a)=b, then f⁻¹(b)=a.
Logarithm: The exponent to which a base must be raised to produce a given number.
Exponential Function: A function where the variable is in the exponent, e.g., y = b^x.
Asymptote: A line that a curve approaches but never touches.
Multiplicity: Number of times a factor appears in a polynomial; affects graph behavior at zeros.
Rational Root Theorem: Helps find possible rational zeros of a polynomial with integer coefficients.

Pro Tips

Domain of f is the range of f⁻¹, and vice versa.
Check for extraneous solutions in log equations (arguments > 0).
Multiplicity of a zero determines if graph crosses (odd) or bounces (even) at x-axis.
Even degree polynomials have same end behavior on both sides.
Odd degree polynomials have opposite end behavior on each side.

Pitfalls to Avoid

Forgetting to check for extraneous solutions in log equations.
Confusing log properties (e.g., log(A+B) ≠ logA + logB).
Incorrectly applying sum/difference of cubes formulas.
Assuming polynomials only have real roots; complex roots appear in conjugate pairs.
Mistaking graph 'bounce' (even multiplicity) for a 'cross' (odd multiplicity).