Functions and Graphs

• This document pertains to Chapter 1: Functions and Their Graphs

Intro to Functions & Their Graphs

• A relation is a connection between variables and values
• Relations are graphically represented as pairs (x, y).
• A function is a special type of relation where each input has at most one output.
• The Vertical Line Test can quickly determine if a graph is a function: if any vertical line passes through more than one point, the graph is not a function
• To verify if an equation is a function, solve for y as a first step.
• If the graph fails the vertical line test, it's not a function
• If any x's result in multiple y's, it is not a function.
• If an equation has an even power of y, then it is not a function
• If an equation is a function, it can be written using function notation that replaces y.
• y = 3x - 4 is the same as f(x) = 3x-4.

Finding The Domain And Range Of A Graph

• The domain of a graph consists of the allowed x-values
• The range of a graph represents the allowed y-values
• To find the domain of a graph, "squish" it to the x-axis
• To find the range of a graph, "squish" it to the y-axis
• [ and ] symbols mean INCLUDE the value
• ≤ and ≥ symbols mean INCLUDE the value.
• The ( and ) symbols mean DON'T INCLUDE the value
• ≤ and ≥ symbols mean DON'T INCLUDE the value.
• When multiple intervals or jumps are present in a graph, use the union symbol to join them.

Finding the Domain of an Equation

• It is at times neccessary to find the domain (allowed x-values) of a function given an equation instead of a graph
• Restriction values can be determined to specify the x-values the function allows.
• Restrictions can be determined by finding the square roots or fractions
• Domain of a square root dictates the x-values that don't make the inside of the square root negative.
• Domain of a fraction dictates the x-values that don't make the denominator 0.

Common Functions

• Graphs of common functions will show up in due course.
• Constant Function: y = same value, it does NOT depend on x.
• Identity Function: f(x) = x
• Shape is a parabola for the Square Function
• (Output = square of input) f(x) = x²
• (Output = cube of input) f(x) = x³
• x CAN be negative, x [CAN | CANT] be negative
• Shape is "L" for a Square Root Function
• Shape is "sideways S" for a Cube Root Function

Transformations

• Transformations occur when a function is modified and changes position and/or shape.
• The transformations of functions include reflection, shift, and stretch

Reflections

• A reflection is a transformation where the function appears to be "mirrored" over the x-axis or the y-axis
• Reflection over the x-axis (y's) changes sign
• Reflection over the y-axis (x's) changes sign

Shifting A function

• A shift occurs when a function is moved vertically and/or horizontally from its original position
• Vertical shifts change the y's.
• Horizontal shifts change the x's.
• For f(x) + k graph shifts up
• For f(x) – k, graph shifts down
• For f(x + h), graph shifts left
• For f(x – h), graph shifts right

Stretches And Shrinks

• Stretches/Shrinks occur when a factor "c" is multiplied inside or outside the function.
• Horizontal Stretch/Compress: f(x) → f(cx)
• Vertical Stretch/Compress: f(x) → c f(x)
• When 0<C<1, [ STRETCHES | SHRINKS ] the graph vertically
• When c > 1, [ STRETCHES | SHRINKS ] the graph vertically
• The same is true for horizontal stretch/compress

Domain & Range of Transformed Functions

• A transformation can affect the domain and range of a function.
• The domain and range of a transformed function can be found by graphing the function

Function Operations

• Functions are added & subtracted just like polynomials.
• Combine like terms
• The domain of f + g or f – g is all the numbers that are common between the domains of f & g
• Domain: Set of common numbers to domains of f & g AND where g(x) ≠.
• Always determine the domain restrictions before simplifying the functions.

Function Composition

• Function composition is like evaluating, but the inside variable of a function is replaced with another function.
• f(g(x)) is often written as
• Evaluating A Function
• f(7) = ()² + 3( ) - 10
• Result is a [ NUMBER | FUNCTION ]
• Composing a function
• f(g(x)) = ( )² + 3( )-10
• Results are [ NUMBER | FUNCTION ]

Evaluating Composed Functions

• It is at times required that the functions are composed and then evaluated at a specific value, f(g(#)). Two common methods are used:
• Method 1: Compose → Evaluate when first asked to find f(g(x))
• Method 2: Evaluate inside → Evaluate outside

Domain Of Composite Functions

• To find the domain of a composite function, follow these steps to exclude x-values from 𝑓 ∘ 𝑔(𝑥):
  • Find any x-values not defined for 𝑓
  • Find any x-values that make 𝑔(𝑥) not defined for 𝑓

Decomposing Functions

• Function decomposition is the reverse of function composition.