Lecture Notes: Functions and Related Concepts

Introduction to Functions

• A function describes the relationship between input (x values) and output (y values).
• Each input is mapped to exactly one output.
• Example: Biological mother function maps a person to their biological mother (one input, one output).

Function Notation

• Functions can be expressed as:
  f(x) = x^2 + 1
• f(2) means substitute 2 for x:
  f(2) = 2^2 + 1 = 5
• For expressions like f(a + 3), substitute the entire expression for x.

Domain and Range

• Domain: Set of all possible x values.
• Range: Set of all possible y values.
• Example function:
  • For g(x) = sqrt(x), domain is x >= 0 (since you can't take the square root of a negative).
  • Range is y >= 0 (outputs are non-negative).

Toolkit Functions

Graphs of Common Functions

1. Linear Function: y = mx + b
   • Straight line, slope (m), y-intercept (b).
2. Quadratic Function: y = ax^2 + bx + c
   • Parabola, opens up/down depending on the sign of a.
3. Exponential Function: y = a * b^x
   • Rapid growth/decay, base greater than 1 indicates growth.
4. Logarithmic Function:
   • The inverse of exponential functions, log properties apply.

Transformations of Functions

• Vertical transformations affect y values; horizontal transformations affect x values.
• For a function f(x):
  • f(x) + k: shifts vertical by k.
  • f(x - h): shifts horizontal by h.
  • a * f(x): stretches (if a > 1) or shrinks (if 0 < a < 1) vertically.

Graphing Quadratic Functions

• Finding Vertex: Use vertex formula x = -b/(2a).
• Use intercepts to sketch the graph.
• Vertex form: y = a(x-h)^2 + k (h,k is the vertex).
• Standard form: y = ax^2 + bx + c.

Exponential Functions

• Form: f(x) = a * b^x, where a is the initial amount, b is the growth factor.
• For continuous growth: P(t) = A * e^(rt).
• Domain for exponential functions is all real numbers; range is positive values.

Logarithmic Functions

• Definition: log_a(b) = c if a^c = b.
• Domain: Positive real numbers only.
• Log rules:
  • Product rule: log_a(xy) = log_a(x) + log_a(y)
  • Quotient rule: log_a(x/y) = log_a(x) - log_a(y)
  • Power rule: log_a(x^n) = n * log_a(x).

Inverse Functions

• Inverses swap the roles of x and y.
• Graphically: The graph of y = f(x) is reflected over the line y = x for its inverse.
• The inverse function exists only if the original function is one-to-one.

Applications of Exponential Functions

1. Population Growth:

   • Model: P(t) = A * (1 + r)^t for growth.
   • P(t) = A * (1 - r)^t for decay.

2. Half-Life Problems:

   • Formula: N(t) = N_0 * (1/2)^(t/T_half).
   • Useful for radioactive decay.

Rational Functions

• Form: f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.
• Asymptotes:
  • Horizontal: determined by degrees of p and q.
  • Vertical: where q(x) = 0.

Conclusion

• Understanding functions, their transformations, and relationships is crucial in algebra, calculus, and real-world applications.