A function \(f\) is a relation between a set of inputs (domain) and a set of possible outputs (range) where each input is related to exactly one output.

Notation: \(f(x) = y\), where \(x\) is the input and \(y\) is the output.

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

A polynomial function is a function that can be expressed in the form \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants and \(n\) is a non-negative integer.

Key Concepts:

• Degree: The highest exponent in the polynomial (e.g., the degree of \(2x^3 + 3x^2 - x + 5\) is 3).
• Leading Coefficient: The coefficient of the term with the highest exponent (e.g., in \(2x^3 + 3x^2 - x + 5\), the leading coefficient is 2).

A rational function is a function that can be expressed as the quotient of two polynomials: \(R(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \neq 0\).

Key Concepts:

• Vertical Asymptotes: Occur where the denominator \(Q(x)\) is zero and the numerator \(P(x)\) is not zero at those points.
• Horizontal Asymptotes: Determined by the degrees of the numerator and denominator.

An exponential function is a function of the form \(f(x) = a \cdot b^x\), where \(a\) and \(b\) are constants, and \(b > 0\) and \(b \neq 1\).

Key Concepts:

• Growth and Decay: If \(b > 1\), the function represents exponential growth. If \(0 < b < 1\), the function represents exponential decay.
• Horizontal Asymptote: The horizontal asymptote of \(f(x) = a \cdot b^x\) is \(y = 0\).

A logarithmic function is the inverse of an exponential function and has the form \(f(x) = \log_b(x)\), where \(b > 0\) and \(b \neq 1\).

Key Concepts:

• Properties of Logarithms:
  • \(\log_b(mn) = \log_b(m) + \log_b(n)\)
  • \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)
  • \(\log_b(m^n) = n \cdot \log_b(m)\)
• Vertical Asymptote: The vertical asymptote of \(f(x) = \log_b(x)\) is \(x = 0\).

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Trigonometric functions can be defined using the unit circle:

• Sine: \( \sin \theta = \text{y-coordinate of the point on the unit circle} \)
• Cosine: \( \cos \theta = \text{X-coordinate of the point on the unit circle} \)
• Tangent: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Trigonometric identities are equations that are true for all values of the variable within their domains. Some important identities include:

• Pythagorean Identities:
  • \( \sin^2 \theta + \cos^2 \theta = 1 \)
  • \( 1 + \tan^2 \theta = \sec^2 \theta \)
  • \( 1 + \cot^2 \theta = \csc^2 \theta \)
• Angle Sum and Difference Identities:
  • \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
  • \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)
  • \( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)
• Double-Angle Identities:
  • \( \sin(2\theta) = 2 \sin \theta \cos \theta \)
  • \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \)
  • \( \tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta} \)

A parabola is the set of all points in a plane equidistant from a given point (focus) and a given line (directrix). The standard form of a parabola is \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\).

Vertex Form: The vertex form of a parabola is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex.

A circle is a set of all points in a plane that are equidistant from a fixed point called the center. The standard form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where (h, k) is the center and \(r\) is the radius.

An ellipse is a set of all points in a plane where the sum of the distances from two fixed points (the foci) is constant. The standard form of an ellipse is \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where (h, k) is the center, \(a\) is the distance from the center to a vertex along the major axis, and \(b\) is the distance from the center to a vertex along the minor axis.

A hyperbola is a set of all points in a plane where the difference of the distances from two fixed points (the foci) is constant. The standard form of a hyperbola is \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\) (horizontal hyperbola) or \(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\) (vertical hyperbola).

Asymptotes: The asymptotes of a hyperbola are the lines that the hyperbola approaches but never touches. For a hyperbola centered at (h, k), the asymptotes are given by the equations \(y - k = \pm \frac{b}{a}(x - h)\) (horizontal hyperbola) or \(y - k = \pm \frac{a}{b}(x - h)\) (vertical hyperbola).

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. The general term of an arithmetic sequence is given by \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference.

Sum of n Terms:

• The sum of the first \(n\) terms of an arithmetic sequence is \(S_n = \frac{n}{2}(a_1 + a_n)\).

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general term of a geometric sequence is given by \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Sum of n Terms:

• The sum of the first \(n\) terms of a geometric sequence is \(S_n = a_1 \frac{1 - r^n}{1 - r}\) for \(r \neq 1\).

An infinite geometric series is a geometric series with an infinite number of terms. It converges to a sum if the common ratio \(r\) satisfies \(|r| < 1\), and the sum is given by:

\[ S = \frac{a_1}{1 - r} \]

An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first \(n\) terms of an arithmetic series is given by:

\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \]

where \(a_1\) is the first term, \(a_n\) is the nth term, and \(n\) is the number of terms.

A geometric series is the sum of the terms of a geometric sequence. The sum of the first \(n\) terms of a geometric series is given by:

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

where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

A vector is a quantity that has both magnitude and direction. Vectors can be represented in the form \( \mathbf{v} = \langle x, y \rangle \) in two dimensions or \( \mathbf{v} = \langle x, y, z \rangle \) in three dimensions.

Key Operations:

• Vector Addition: \( \mathbf{u} + \mathbf{v} = \langle u_x + v_x, u_y + v_y \rangle \)
• Scalar Multiplication: \( c\mathbf{v} = \langle cx, cy \rangle \)
• Dot Product: \( \mathbf{u} \cdot \mathbf{v} = u_xv_x + u_yv_y \)
• Cross Product: \( \mathbf{u} \times \mathbf{v} = \langle u_yv_z - u_zv_y, u_zv_x - u_xv_z, u_xv_y - u_yv_x \rangle \) (for 3D vectors)

The magnitude (length) of a vector \( \mathbf{v} = \langle x, y \rangle \) is given by:

\[ |\mathbf{v}| = \sqrt{x^2 + y^2} \]

The direction of a vector is given by the angle \( \theta \) it makes with the positive x-axis, calculated as:

\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

The projection of vector \( \mathbf{u} \) onto vector \( \mathbf{v} \) is given by:

\[ \text{Proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \mathbf{v} \]

A matrix is a rectangular array of numbers arranged in rows and columns. The most common matrix operations include addition, subtraction, and multiplication.

Key Operations:

• Addition and Subtraction: Matrices can be added or subtracted if they have the same dimensions. The operation is performed element-wise.
• Multiplication: The product of two matrices \(A\) (of size \(m \times n\)) and \(B\) (of size \(n \times p\)) is a new matrix \(C\) of size \(m \times p\), where each element of \(C\) is the dot product of the corresponding row of \(A\) and the column of \(B\).
• Transpose: The transpose of a matrix \(A\) is obtained by flipping it over its diagonal, resulting in a new matrix \(A^T\) where the rows become columns and the columns become rows.

The determinant of a square matrix \(A\) is a scalar value that can be computed from its elements and provides important properties of the matrix. The inverse of a square matrix \(A\) is a matrix \(A^{-1}\) such that \(AA^{-1} = I\), where \(I\) is the identity matrix.

Key Formulas:

• Determinant of a 2x2 Matrix: \[ \text{det}(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \]
• Inverse of a 2x2 Matrix: \[ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

A complex number is a number of the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with the property \(i^2 = -1\).

Key Operations:

• Addition and Subtraction: \( (a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i \)
• Multiplication: \( (a + bi)(c + di) = (ac - bd) + (ad + bc)i \)
• Conjugate: The conjugate of \(a + bi\) is \(a - bi\).
• Division: To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator.

Complex numbers can also be represented in polar form as \( r(\cos \theta + i \sin \theta) \), where \( r = \sqrt{a^2 + b^2} \) is the magnitude and \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \) is the argument (angle).

The polar form is often written as \( r \text{cis} \theta \), where \( \text{cis} \theta = \cos \theta + i \sin \theta \).

De Moivre's Theorem states that for any complex number in polar form \( r(\cos \theta + i \sin \theta) \) and any integer \( n \),

\[ [r(\cos \theta + i \sin \theta)]^n = r^n (\cos n\theta + i \sin n\theta) \]

This theorem is useful for finding powers and roots of complex numbers.