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Pre-Calculus Parent Functions

Sep 4, 2025

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Overview

This lecture covers key parent functions in pre-calculus, their graphs, domains and ranges, and discusses function transformations, composition, and inverses.

Parent Functions and Their Properties

  • Linear Function (y = x): Graph is a line through the origin; domain and range are all real numbers (βˆ’βˆž, ∞).
  • Quadratic Function (y = xΒ²): Parabola opening upward; domain (βˆ’βˆž, ∞), range [0, ∞).
  • Cubic Function (y = xΒ³): S-shaped curve; domain and range are (βˆ’βˆž, ∞).
  • Square Root Function (y = √x): Starts at (0,0) and increases; domain [0, ∞), range [0, ∞).
  • Cube Root Function (y = ³√x): S-shaped, symmetric about origin; domain and range (βˆ’βˆž, ∞).
  • Absolute Value Function (y = |x|): V-shaped, opens up; domain (βˆ’βˆž, ∞), range [0, ∞).
  • Rational Function (y = 1/x): Two branches, vertical asymptote at x=0, horizontal at y=0; domain (βˆ’βˆž, 0) βˆͺ (0, ∞), range (βˆ’βˆž, 0) βˆͺ (0, ∞).
  • Rational Function (y = 1/xΒ²): Symmetric branches above x-axis; domain (βˆ’βˆž, 0) βˆͺ (0, ∞), range (0, ∞).
  • Exponential Function (y = eΛ£): Rapid increase, horizontal asymptote at y=0; domain (βˆ’βˆž, ∞), range (0, ∞).
  • Natural Log Function (y = ln x): Vertical asymptote at x=0, increases slowly; domain (0, ∞), range (βˆ’βˆž, ∞).
  • Sine and Cosine Functions (y = sin x, y = cos x): Periodic waves; domain (βˆ’βˆž, ∞), range [βˆ’1, 1].
  • Tangent Function (y = tan x): Vertical asymptotes at odd multiples of Ο€/2; domain excludes these, range (βˆ’βˆž, ∞).

Function Transformations

  • Vertical Stretch/Shrink: f(x) β†’ aΒ·f(x) stretches by |a| if a>1, shrinks if 0<a<1.
  • Horizontal Stretch/Shrink: f(x) β†’ f(bx); b>1 shrinks, 0<b<1 stretches horizontally.
  • Vertical Shift: f(x) + c shifts up by c; f(x) βˆ’ c shifts down by c.
  • Horizontal Shift: f(xβˆ’d) shifts right d units; f(x+d) shifts left d units.
  • Reflections: βˆ’f(x) reflects over x-axis; f(βˆ’x) reflects over y-axis; both reflect over the origin.
  • Combined Transformations: Apply shifts, stretches, and reflections as indicated by multiple modifications.

Compositions and Inverses of Functions

  • Composition (f ∘ g)(x): Plug g(x) into f(x); simplify as needed.
  • Inverse Function: Swap x and y, solve for y; f and f⁻¹ are inverses if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Determining Domain and Range with Transformations

  • **Shifts and stretches affect domain and range based on the parent function’s properties and the transformation’s direction/magnitude.
  • **For rational functions, remove values where the denominator is zero (vertical asymptotes) from the domain and horizontal asymptotes from the range.

Key Terms & Definitions

  • Domain β€” set of all possible input (x) values for a function.
  • Range β€” set of all possible output (y) values for a function.
  • Parent Function β€” the simplest form of a function type, used as a base for transformations.
  • Asymptote β€” a line that a graph approaches but never touches.
  • Amplitude β€” the height from the center line to the peak (for trig functions).
  • Composite Function β€” a function created by plugging one function into another.
  • Inverse Function β€” a function that reverses the effect of the original function.

Action Items / Next Steps

  • Practice graphing each parent function and applying transformations.
  • Complete assigned problems on domain, range, and inverse/composite functions.
  • Review any recommended textbook sections on functions and transformations.

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