Functions Flashcards
Mapping,
meaning
to be a function
A function is a rule of correspondence where each element of one set x is assigned to exactly one element of another set y
To be a function for any given x value there must be only one y value, also known as mapping
x –> y
Ordered Pairs,
4 types with examples and graphs
One to one, linear
eg. y = x + 1
One to many, horizontal parabola
Many to one, parabola/quadratic
eg. y = x^2
Many to many, circular (not a function)
Notation,
dependance of variables
function notation
The the element of the first set of values are x and the element of the second set of values are y then variable x is independent and variable y is dependant
If the function is named f and the independent variable is x, then the function notation f(x) can be used instead of y.
Types of numbers,
natural numbers
whole numbers
integers
real numbers
visual representation of number types
Counting numbers, excluding zero, negatives and decimals
ℕ = 1, 2, 3, 4, 5, …
All the natural numbers including zero
\W = 0, 1, 2, 3, 4, …
All the whole numbers and their negatives too
ℤ = …, -2, -1, 0, 1, 2, …
Has all numbers commonly dealt with including, positives, negatives and decimals (ℝ)
ℕ ) \W) ℤ) ℝ
Domain and range,
domain meaning
range meaning
determinant of function similarity
example of domain and range for a function
The set of all possible values of the independent variable x is called the domain of the function,
how far the x values go in the x-axis direction
The corresponding set of all possible values of the dependant variable y is called the range of the function,
how far the y values go in the y-axis direction
The domain is an essential part of a function, if 2 functions f(x) and g(x) have different domains, they are different functions, even if they have the same rule
List domain and range for f(x) = x^2 + 1 (Graph sketched):
D: x ∈ ℝ
R: y ≥ 1, y ∈ ℝ
Note: “∈” means belongs too
Hyperbola,
formula
curve composition
horizontal asymptote process
vertical asymptote process
basic shapes of a hyperbola
exclusions in domain and range of a function / example
(ax + b) / (cx + d) where cx + d ≠ 0
Hyperbolic curves have 2 asymptotes, vertical and horizontal
Let x = ∞
y = (ax + b) / (cx + d) = (a∞ + b) / (c∞ + d)
y = a / c
Let the denominator = 0
cx + d = 0
cx = -d
x = -d / c
Positive:
curve that gets close to the axis in negative, negative quadrant of the graph
curve that gets close to the axis in positive, positive quadrant of the graph
Negative:
curve that gets close to the axis in both positive, negative quadrants of the graph
Division by zero is not defined (k/0 is undefined), thus any value of x that creates division by zero can not be in the domain of a function
List the domain and range of y = (2x - 3) / (3- x)
D: x ≠ 3, x ∈ ℝ
R: y ≠ -2, y ∈ ℝ
Graph of square root functions,
formula
function rule
y = √ax + b where ax + b ≠ to a negative number
Square roots of negative numbers are not real numbers thus any value of x creating the square root of a negative number can not be in the domain of a function.
Evaluate functions,
description
example
Given a function, its numerical value depends on the value of x
f(x) = 3x + 2 x = 2 f(x) = 3(2) + 2 f(x) = 8
Composite functions,
meaning
types
combined function, given a list of coordinates for 2
This means a function of a function, we can combine 2 or more functions to give a composite function
fg(x) means g(x) goes into f gf(x) means f(x) goes into g hence gf(x) ≠ fg(x) ff(x) = f^2(x)
The combined function would have the first element, or x element, of the function closest to x, as the first coordinate
Inverse functions,
description
notation
range and domain in terms of f and f^-1
ordered pairs with an inverse
composite functions with inverse rule
finding the inverse functions
If a function f maps x to y then its inverse would map y to x, undoes the function effect
f^-1(x)
The range of f is the domain of f^-1 and the domain of f is the range of f^-1
f^-1 only exists if f is a one to one function, as if f is a many to one function the inverse is not an inverse function instead it is a one to many inverse relation
ff^-1(x) = f^-1f(x) = x
(f^-1)^-1(x) = f(x)
(fg)^-1 = g^-1 f^-1
f^-1(x) ≠ 1 / (f(x))
Change the notation of f(x) to y and swap the position of the x and y variable, then rearrange the equation to make x the subject, finally swap x and y once again and change the notation to y = f^-1(x)
Graph of an Inverse function,
description
If the graph of a function y = f(x) is given the graph of its inverse function y = f^-1(x) is the reflection of y = f(x) in the line y = x
Graphs of absolute value functions,
description
absolute value function defined (formula)
linear equations;
formula
process
non-linear equations;
formula
process
The absolute value of a function f(x) is a function which has the same numerical value for f(x) for all values of x and is denoted by | f(x) |
Linear equations:
y = | mx + c | , a ≤ x ≤ b
Find the vertex (where y = 0), x = -c / m
Find the 2 extreme points, where x = a and x = b
Put x = a into y = | mx + c | and solve for y .. (a, y1)
Non-linear equations:
y = ax^2 + bx + c, a ≤ x ≤ b
Sketch the graph of y = f(x)
Reflect the part of the graph below the x-axis up, y ≥ 0
f(x) | = …
f(x) f(x) > 0
0 f(x) = 0
-f(x) f(x) < 0
Transformations of functions,
translations; formula description translation vector process rule/formula of vector translation
reflections; formula description types of reflections reflection rule/formula
stretches; formula description types of stretches negative stretch
Translations:
y = f(x ± a) and y = f(x) ± b
In translation every point moves the same distance, when describing this it is necessary to give a translation vector
Translation vector (a b) {a written over b vertically}
y = f(x) ± b is to translate the graph y = f(x) vertically,
[y-axis], through b units, b > 0 moves upwards and b < 0 moves downwards
y = f(x ± a) is to translate the graph of y = f(x) horizontally, [x-axis], through a units, a > 0 moves to the right and a < 0 moves to the left
The graph of y = f(x ± a) ± b is a translation of the graph
y = f(x) by the vector (∓a ±b)
Reflections:
y = -f(x) and y = f(-x)
In reflection there is an axis of symmetry or mirror line, the image A’ of a point A is on the opposite side of the line to the object A, but the same distance
y = -f(x) is to reflect the graph y = f(x), [in the x-axis], up or down
y = f(-x) is to reflect the graph y = f(x), [in the y-axis], left or right
The graph of y = -f(-x) is a reflection of the graph y = f(x) in the x-axis and y-axis.
Stretches:
y = pf(x), p > 0 and y = f(qx), q > 0
If a function undergoes a stretch the effect is a lengthening in one direction only. When describing a stretch 2 pieces of information need to be given, the scale factor and the invariant line.
y = pf(x), p > 0 is to vertically stretch (parallel to the y-axis) the graph y = f(x) by the scale factor p,
- If p > 1 it moves point of y = f(x) further away from the x-axis
- If p > 0 and p < 1 it moves points of y = f(x) closer to the x-axis
y = f(qx), q > 0 is to horizontally stretch (parallel to the x-axis) the graph y = f(x) by the scale factor 1/q
- If q > 1 it moves points of y = f(x) closer to the y-axis
- If q > 0 and q < 1 it moves points of y = f(x) further away from the y-axis
If a is lesser than 0 then y = af(x) can be considered a stretch of y = f(x) with a negative scale factor or as a stretch with positive scale factor followed by a reflection in the x-axis.
Combined transformations,
summary;
vertical transformations
horizontal transformations
combined different transformations note
combined similar transformations note
2 vertical transformations
2 horizontal transformations
Vertical transformations:
- y = f(x) ± a –> translation (0 ± a)
- y = -f(x) –> reflection in x-axis
- y = af(x) –> vertical stretch (parallel to the y-axis), factor a
Horizontal transformations:
- y = f(x ± a) –> translation (∓ a 0)
- y = f(-x) –> reflection in y-axis
- y = f(ax) –> horizontal stretch (parallel to x-axis), factor 1/a
When one horizontal and one vertical transformation are applied the order in which they are applied does not affect the outcome
When 2 vertical or 2 horizontal transformations are combined the order in which they are applied may affect the outcome
Vertical transformations follow the normal order of operations, as used in arithmetic,
y = af(x) + k
1. stretch vertically, factor a
2. translate vertically constant k
(multiply by function a then add constant k)
Horizontal transformations follow the opposite order of operations, as used in arithmetic,
y = f(bx + c)
1. translate horizontally, constant -c
2. stretch horizontally, factor 1/b
(replace x with x + c then replace x with bx)
