Core Concepts of Functions

Overview

This lecture covers the core concepts of functions in pre-calculus, including types of parent functions, their graphs, domain and range, function transformations, inverse and composite functions, and how these features appear for various function types.

Parent Functions: Graphs, Domain, and Range

• Linear function (y = x): domain and range are all real numbers (-∞, ∞).
• Quadratic function (y = x²): domain is all real numbers, range is [0, ∞).
• Cubic function (y = x³): domain and range are all real numbers.
• Square root function (y = √x): domain and range are [0, ∞).
• Cube root function (y = ³√x): domain and range are all real numbers.
• Absolute value function (y = |x|): domain is all real numbers, range is [0, ∞).
• Rational function (y = 1/x): domain and range are (-∞, 0) ∪ (0, ∞), with asymptotes at x=0 and y=0.
• Rational function (y = 1/x²): domain is (-∞, 0) ∪ (0, ∞), range is (0, ∞), asymptotes at x=0 and y=0.
• Exponential function (y = eˣ): domain is all real numbers, range is (0, ∞).
• Natural log function (y = ln x): domain is (0, ∞), range is all real numbers.
• Sine and cosine functions: domain is all real numbers, range is [-1, 1] (or adjusted by amplitude).
• Tangent function: domain is all reals except x ≠ (2n+1)π/2, range is all real numbers.

Function Transformations

• Vertical stretch: multiply by a factor outside the function (e.g., 2f(x)).
• Vertical shrink: multiply by a fraction outside the function (e.g., (1/2)f(x)).
• Horizontal shrink: multiply x by a factor inside the function (e.g., f(2x)).
• Horizontal stretch: multiply x by a fraction inside the function (e.g., f(x/2)).
• Vertical shift: add/subtract a constant outside (e.g., f(x) + 3 shifts up).
• Horizontal shift: add/subtract inside (e.g., f(x - 4) shifts right).
• Reflect over x-axis: multiply by -1 outside (e.g., -f(x)).
• Reflect over y-axis: multiply x by -1 inside (e.g., f(-x)).
• Reflection over the origin: combine both reflections.

Inverse and Composite Functions

• To find an inverse, swap x and y and solve for y.
• Two functions are inverses if f(g(x)) = x and g(f(x)) = x.
• Composite function f(g(x)): insert g(x) into f(x).
• To evaluate composites at a number, first evaluate the inner function, use its output within the outer function.

Graphing Transformed and Special Functions

• Adjust domain, range, and asymptotes based on shifts and reflections.
• For vertical/horizontal shifts, move asymptotes and critical points accordingly.
• For root and absolute value functions, note direction and new origins from shifts.

Key Terms & Definitions

• Domain — all possible input (x) values for a function.
• Range — all possible output (y) values for a function.
• Parent function — the simplest form of a function type.
• Asymptote — a line the graph approaches but never touches.
• Amplitude — the maximum value from the centerline in trig functions.
• Composite function — a function formed by applying one function to the result of another.
• Inverse function — a function that reverses another function’s effect.

Action Items / Next Steps

• Practice identifying parent functions and graph their domain and range.
• Complete exercises on function transformations and their effects.
• Work through example problems on composite and inverse functions.
• Review graphing with asymptotes, especially for rational and exponential functions.