Equations and Inequalities

Solving Exponential and Logarithmic Equations

Exponential and logarithmic equations are fundamental in precalculus, often requiring algebraic manipulation and properties of exponents and logarithms.

• Exponential Equation: An equation where the variable appears in the exponent, e.g., .

• Logarithmic Equation: An equation involving logarithms, e.g., .

• Solving Strategy: Use properties such as and .

• Example: Solve . Rewrite as .

Functions & Graphs

Domain and Range

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

• Example: For , the domain is all real numbers except ; the range is all real numbers except .

• Horizontal Asymptote: For rational functions, the horizontal asymptote can be found by comparing degrees of numerator and denominator.

• Example: Find the horizontal asymptote of . As degrees are equal, asymptote is .

Polynomial and Rational Functions

Quadratic Equations and Roots

Quadratic equations are polynomials of degree 2 and can have real or complex roots.

• General Form:

• Number of Real Roots: Determined by the discriminant .

• Example: has discriminant $0$, so one real root.

Exponential and Logarithmic Functions

Properties and Applications

Exponential functions model growth and decay, while logarithmic functions are their inverses.

• Exponential Growth:

• Compound Interest:

• Logarithmic Properties:

• Example: If $1200 compounded quarterly, use to find future value.

Graphs

Transformations of Functions

Transformations include shifts, stretches, and reflections of graphs.

• Vertical Shift: shifts up/down.

• Horizontal Shift: shifts right/left.

• Reflection: reflects over x-axis.

• Example: is a parabola shifted right 4 units and up 10 units.

Conic Sections

Circles and Parabolas

Conic sections include circles, ellipses, parabolas, and hyperbolas, each with standard equations.

• Circle:

• Parabola:

• Example: is a parabola with vertex at .

Sequences and Series

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers, and series are their sums.

• Arithmetic Sequence:

• Geometric Sequence:

• Example: Find the 7th term of : .

Functions: Inverses and Composition

Inverse Functions

The inverse of a function reverses its input and output.

• Notation:

• Finding Inverse: Swap and and solve for .

• Example: If , then .

Contest Problems: Logarithmic and Exponential Graphs

Domain of Logarithmic Functions

The domain of is .

• Example: For , domain is .

Graphing Exponential Functions

Exponential functions have characteristic graphs: increasing for , decreasing for .

• Example: If , then graphs as a shifted exponential curve.

Summary Table: Key Properties of Functions

Additional info: Some explanations and examples were expanded for clarity and completeness based on standard precalculus curriculum.