What is a one-to-one mapping?

A one-to-one mapping is a transformation where each input is mapped to exactly one unique output , and no two inputs are mapped to the same output .

True or False? A many-to-one mapping can be a function .

True. A many-to-one mapping can be a function .

What is the vertical line test ?

The vertical line test is a method to determine if a graph represents a function . If a graph does represent a function , then any vertical line should intersect with the graph once at most .

What is the domain of a function?

The domain of a function is the set of values that are used as inputs for the function.

What is the range of a function?

The range of a function is the set of values that are given as outputs by the function.

True or False? An inverse function reverses the effect of a function .

True. An inverse function reverses the effect of a function .

True or False? Only a many-to-one function has an inverse function.

False. A many-to-one function does not have an inverse function. Only one-to-one functions have inverses .

What is the horizontal line test ?

The horizontal line test is a method to determine if a function has an inverse . If a function has an inverse , then any horizontal line should intersect with the graph once at most .

What is the relationship between the domain of a function and its inverse function ?

The domain of a function becomes the range of its inverse function .

True or False? For a many-to-one function with a restricted domain , the range of its inverse function is the same as the original function 's domain .

False. For a many-to-one function with a restricted domain, the range of its inverse function is the not the same as the original function's domain. The range of the inverse function is the same as the restricted domain of the original function .

True or False? Some trig functions are even functions.

True. Some trig functions are even functions, e.g. cos x and sec x .

True or False? The graph of an even function has rotational symmetry .

False. The graph of an even function does not have rotational symmetry . The graph of an odd function has rotational symmetry .

What is the difference between the command terms ' draw ' and ' sketch ' when graphing?

To sketch : show the general shape of the graph and label key points and axes. To draw : use a pencil and ruler to draw the graph to scale, plot points accurately, join points with a smooth curve or a straight line, and label key points and axes.

An asymptote is a line which the graph will get closer and closer to but not touch .

True or False? Most GDC models will automatically plot asymptotes .

False. Most GDC makes/models will not plot or show asymptotes just from inputting a function.

Define local minimum ( maximum ).

A local minimum ( maximum ) is a point at which the graph reaches the minimum ( maximum ) value that it takes in the immediate vicinity of the point. The graph may reach lower (higher) values further away from the point. It is also called a turning point .

True or False? A local minimum / maximum is always the global minimum / maximum of a function.

False. A local minimum / maximum is not necessarily the global minimum / maximum (the minimum/maximum of the whole graph).

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What is a one-to-one mapping?
A one-to-one mapping is a transformation where each input is mapped to exactly one unique output, and no two inputs are mapped to the same output.
True or False?
A many-to-one mapping can be a function.
True.
A many-to-one mapping can be a function.
What is the vertical line test?
The vertical line test is a method to determine if a graph represents a function.
If a graph does represent a function, then any vertical line should intersect with the graph once at most.
What is the domain of a function?
The domain of a function is the set of values that are used as inputs for the function.
What is the range of a function?
The range of a function is the set of values that are given as outputs by the function.
What symbol is used to represent the set of all real numbers?
$ℝ$ represents the set of all real numbers (i.e. all the numbers that can be placed on a number line).
What does $ℚ$ represent?
$ℚ$ represents the set of all rational numbers, $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .
What is a composite function?
A composite function is a function made up of two or more functions. One function is applied to another function, e.g. $(f \circ g) (x)$ .
True or False?
The order in which the functions in a composite function are applied does not matter.
False.
The order in which the functions in a composite function are applied does matter. $(f \circ g) (x)$ is not usually equivalent to $(g \circ f) (x)$ .
Always apply the function closest to the variable first.
E.g. for the composite function $(f \circ g) (x)$ , you must first apply the function $g$ followed by the function $f$ .
To find the domain and range of a composite function, what is the first step you need to take?
To find the domain and range of a composite function, the first step is you need to take is to find the range of the first function that is applied.
E.g. To find the domain and range for the function $(f \circ g) (x)$ , you must first find the range of $g$ .
True or False?
An inverse function reverses the effect of a function.
True.
An inverse function reverses the effect of a function.
True or False?
Only a many-to-one function has an inverse function.
False.
A many-to-one function does not have an inverse function.
Only one-to-one functions have inverses.
What is the horizontal line test?
The horizontal line test is a method to determine if a function has an inverse.
If a function has an inverse, then any horizontal line should intersect with the graph once at most.
What is the relationship between the domain of a function and its inverse function?
The domain of a function becomes the range of its inverse function.
State the relationship between the graphs of $f (x)$ and $f^{- 1} (x)$ .
The graph $y = f^{- 1} (x)$ is a reflection of the graph $y = f (x)$ in the line $y = x$ .
What is the identity function?
The identity function maps each value in a function to itself.
$id (x) = x$ .
If two composite functions, $f \circ g$ and $g \circ f$ , have the same effect as the identity function, what can be said about the functions $f$ and $g$ ?
If two composite functions, $f \circ g$ and $g \circ f$ , have the same effect as the identity function, then the functions $f$ and $g$ must be inverse functions of each other.
How can you adapt a many-to-one function in order for it to have an inverse?
You can restrict the domain of a many-to-one function to a subset of the domain on which the function is one-to-one. Then the function on the restricted domain will have an inverse.
E.g. the function $y = x^{2}$ is a many-to-one function, but if the domain is restricted to $x \geq 0$ , the function becomes one-to-one and has an inverse.
How should the domain for the sine function be restricted in order for it to have an inverse?
In order for the sine function to have an inverse, the domain must be restricted to half a cycle, from a maximum to a minimum or vice versa.
E.g. $- \frac{π}{2} \leq x \leq \frac{π}{2}$ .
True or False?
For a many-to-one function with a restricted domain, the range of its inverse function is the same as the original function's domain.
False.
For a many-to-one function with a restricted domain, the range of its inverse function is the not the same as the original function's domain.
The range of the inverse function is the same as the restricted domain of the original function.
What is an odd function?
An odd function is one where $f (- x) = - f (x)$ for all values of $x$ .
I.e., a horizontal reflection of a function is the same as its vertical reflection.
What must be true about the exponent of a power function if the function is odd?
For a power function to be odd, its exponent must be odd.
E.g. $y = x^{3}$ is an odd function as $y = {(- x)}^{3} = - x^{3}$ but $y = x^{2}$ is not as $y = {(- x)}^{2} \neq - x^{2}$ .
True or False?
cos x is an odd function.
False.
cos x is not an odd function as $- \cos (x) \neq \cos (- x)$ . cos x is an even function.
The other two basic trig functions, sin x and tan x, are odd functions.
What is an even function?
An even function is one where $f (- x) = f (x)$ for all values of $x$ .
I.e., the original function does not change when reflected horizontally (about the y-axis).
True or False?
Some trig functions are even functions.
True.
Some trig functions are even functions, e.g. cos x and sec x.
Is the modulus function, $| x |$ , odd or even?
The modulus function, $| x |$ , is an even function.
True or False?
The graph of an even function has rotational symmetry.
False.
The graph of an even function does not have rotational symmetry.
The graph of an odd function has rotational symmetry.
True or False?
The graph of an even function has reflective symmetry.
True.
The graph of an even function has reflective symmetry.
What is a periodic function?
A periodic function is one where $f (x + k) = f (x)$ for all values of $x$ , where $k$ is the period.
I.e. the graph repeats itself at an interval of $k$ .
True or False?
The graph of a periodic function has translational symmetry.
True.
The graph of a periodic function has translational symmetry.
What is a self-inverse function?
A self-inverse function is one where:
- $(f \circ f) (x) = x$ for all values of $x$
- $f^{- 1} (x) = f (x)$
I.e., a self-inverse function is the same as its inverse.
What is the equation of the line about which a self-inverse function has reflective symmetry.
A self-inverse function has reflective symmetry about the line $y = x$ .
How do you determine if a point $(a, b)$ lies on the graph $y = f (x)$ ?
A point $(a, b)$ lies on the graph $y = f (x)$ if $f (a) = b$ .
What is the difference between the command terms 'draw' and 'sketch' when graphing?
To sketch: show the general shape of the graph and label key points and axes.
To draw: use a pencil and ruler to draw the graph to scale, plot points accurately, join points with a smooth curve or a straight line, and label key points and axes.
Define asymptote.
An asymptote is a line which the graph will get closer and closer to but not touch.
True or False?
Most GDC models will automatically plot asymptotes.
False.
Most GDC makes/models will not plot or show asymptotes just from inputting a function.
How can you use graphs to solve the equation $f (x) = a$ ?
Plot the graphs $y = f (x)$ and $y = a$ on your GDC, and find the points of intersection.
The x-coordinates are the solutions of the equation.
Define local minimum (maximum).
A local minimum (maximum) is a point at which the graph reaches the minimum (maximum) value that it takes in the immediate vicinity of the point. The graph may reach lower (higher) values further away from the point.
It is also called a turning point.
True or False?
A local minimum/maximum is always the global minimum/maximum of a function.
False.
A local minimum/maximum is not necessarily the global minimum/maximum (the minimum/maximum of the whole graph).

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Functions Toolkit (DP IB Analysis & Approaches (AA))

Flashcards

Cards in this collection (39)

1. Number & Algebra

Number & Algebra Toolkit

Exponentials & Logs

Sequences & Series

Simple Proof & Reasoning

Proof by Induction & Contradiction

Binomial Theorem

Permutations & Combinations

Complex Numbers

Further Complex Numbers

Systems of Linear Equations

2. Functions

Linear Functions & Graphs

Quadratic Functions & Graphs

Functions Toolkit

Other Functions & Graphs

Reciprocal & Rational Functions

Transformations of Graphs

Polynomial Functions

Inequalities

Modulus Functions & Further Transformations

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

The Unit Circle & Exact Values

Trigonometric Functions & Graphs

Trigonometric Equations & Identities

Inverse & Reciprocal Trigonometric Functions

Trigonometric Proof & Equation Strategies

Vector Properties

Vector Equations of Lines

Vector Planes

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Probability

Discrete Random Variables

Binomial Distribution

Normal Distribution

Continuous Random Variables

5. Calculus

Differentiation

Techniques & Applications of Differentiation

Integration

Techniques & Applications of Integration

Optimisation

Kinematics

Basic Limits & Continuity

Further Differentiation

Further Integration

Differential Equations

Maclaurin Series

Limits using l'Hôpital's Rule & Maclaurin Series