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.

Test your knowledge of functions and graphs. Topics include function notation, domains, coordinate planes, and reflections. See how functions change with varying inputs and how they are visually represented.