Overview
This lecture covers essential concepts about functions, including types of parent functions, their graphs, domains and ranges, function transformations, composite functions, and inverse functionsβkey foundations for pre-calculus.
Parent Functions and Their Properties
- Linear function ( y = x ): graph is a line through the origin; domain and range are all real numbers ((-\infty, \infty)).
- Quadratic function ( y = x^2 ): upward-opening parabola; domain is all real numbers; range is ([0, \infty)).
- Cubic function ( y = x^3 ): S-shaped curve; domain and range are all real numbers.
- Square root function ( y = \sqrt{x} ): starts at (0,0), only in the first quadrant; domain and range are ([0, \infty)).
- Cube root function ( y = \sqrt[3]{x} ): passes through the origin and extends in all directions; domain and range are all real numbers.
- Absolute value function ( y = |x| ): V-shaped, opens upward; domain is all real numbers; range is ([0, \infty)).
- Rational function ( y = \frac{1}{x} ): vertical asymptote at ( x = 0 ), horizontal at ( y = 0 ); domain and range exclude zero.
- Rational function ( y = \frac{1}{x^2} ): only positive y-values, vertical asymptote at ( x = 0 ), horizontal at ( y = 0 ); domain excludes zero, range is ( (0, \infty) ).
- Exponential function ( y = e^x ): horizontal asymptote at ( y = 0 ); domain is all real numbers, range is ( (0, \infty) ).
- Logarithmic function ( y = \ln x ): vertical asymptote at ( x = 0 ); domain is ( (0, \infty) ), range is all real numbers.
- Sine and cosine functions: periodic waves; domain is all real numbers; range is ([-1, 1]).
- Tangent function: periodic, with vertical asymptotes at ( x = \frac{(2n+1)\pi}{2} ); domain excludes these asymptotes; range is all real numbers.
Transformations of Functions
- Multiplying outside by a constant stretches (( a > 1 )) or shrinks (( 0 < a < 1 )) vertically.
- Multiplying inside by a constant shrinks (( b > 1 )) or stretches (( 0 < b < 1 )) horizontally.
- Adding/subtracting inside shifts horizontally (right if negative, left if positive).
- Adding/subtracting outside shifts vertically (up if positive, down if negative).
- Negative outside reflects over the x-axis; negative inside reflects over the y-axis.
- Both negatives reflect over the origin.
- Inverse function switches x and y values; the graph reflects over ( y = x ).
Finding Domains and Ranges with Transformations
- Identify asymptotes and exclude their values from the domain/range.
- Vertical shifts change the range/asymptote; horizontal shifts change the domain/asymptote.
- Reflections affect which quadrants the graph occupies.
Composite and Inverse Functions
- Composite function ( f(g(x)) ): Plug ( g(x) ) into ( f(x) ); simplify if possible.
- For inverse functions, swap x and y in the equation and solve for y.
- Two functions are inverses if both ( f(g(x)) = x ) and ( g(f(x)) = x ).
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.
- Asymptote β a line that the graph approaches but never touches.
- Transformation β changes in position or shape of the graph (shifts, stretches, reflections).
- Composite function β a function made by plugging one function into another.
- Inverse function β a function that "undoes" another function, reflecting across ( y = x ).
Action Items / Next Steps
- Practice graphing parent and transformed functions.
- Find domains and ranges for given functions and their transformations.
- Work on composition and inversion of functions as shown in examples.
- Prepare for problems involving identifying and applying function transformations.