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Inverse functions and graphs
SLIDE 1
Functions
A function is a map or mapping that takes an input number and computers output number. Functions are generally defined by a rule that tells one what to do to the input, called the argument, to obtain the output, called its value.
function mapping graph
The above are all statements of the same information.
In and in , the letter x represents the input argument, and the letter y represents the output argument. The letters stand for variables, since x may be replaced by different inputs.
For a function such as , where the rule is given, the function is defined by the rule and not by the choice of letters for variables, which is arbitrary. For example, and are the same function.
SLIDE 2
In the following list there are four pairs of identical function. State which function goes with which.
? ?
? ?
? ?
? ?
SLIDE 3
? is the same function as ?
? is the same function as ?
? is the same function as ?
? is the same as ?
SLIDE 4
A function takes one from one number to another. Using a mapping diagram, we can write
“f takes x to y”.
The function that reverses the process, for given input and output, is called the inverse function of f. It is denoted .
SLIDE 5
This is a duplicate of slide 8 from the 7.2 workout
Example
A function is defined by . Make y the subject of and hence find the inverse of .
Solution
Notes to this solution
In the last step we replace the variable y by the original variable x. This is a standard device required in most examinations to demonstrate understanding that the two functions and are one and the same function. The same result can be obtained by switching the variables at the beginning and writing as the line after .
SLIDE 6
Find the inverse of
? ?
SLIDE 7
? ?
SLIDE 8
Every function must be well-defined
Consider the following rule
The function is such that for argument , is sometimes 2 and sometimes 3.
? Why cannot this rule be used to define a function?
SLIDE 9
The function is such that for argument , is sometimes 2 and sometimes 3.
This cannot be used as a definition of a well-defined function, because the person computing the function must always know what the output value is for any given value.
? This means that for any given input there can be only one output.
SLIDE 10
? Solve the equation
? Why is not a well-defined function?
SLIDE 11
? The solution to the equation is .
That is, or .
? Thus, is not a well-defined function, because there are two values of y for any one given value of x.
SLIDE 12
Well-defined function
A function is well-defined if, and only if, for every input argument x there is only one output argument y.
Examples
? is a well-defined function. There is only one value y for every value x.
? is a well-defined function. For every one input argument x, there is only one output argument y. For example, if or the output is determined as .
? is not well-defined, because for every input argument x, there are two possible output values y, so the output is ambiguous and not determined.
SLIDE 13
Which of the following is a well-defined function?
? ?
? ?
Hint. The last example is the inverse sine and is used to find the angle from a given trigonometric ratio. However, you should consider examples such as and sin and the question, what is ?
SLIDE 14
?
? . Every cube root must be positive. If we cube a negative number we get a negative number
? . For example, , so the output is not determined.
? . Following the hint, we find , so when we compute the output is not determined – that is, is the output 30° or is it 150° (or is it something else, for example, 390°).
SLIDE 15
Consider the following definitions
? Why is (1) not well-defined?
? What is the similarity, and what is the difference between and ?
? Assuming that is well-defined whereas is not, what improvement in the definition of has made it well-defined?
SLIDE 16
? is not well-defined, because there is a singularity as . That is, when the function has no value, because we are not allowed to divide by 0.
? and have the same rule, which is , but we have added to the restriction that .
? In the definition of we have improved upon the definition of by adding the restriction to turn the not well-defined into a well-defined function, .
SLIDE 17
Domain
The domain of a function is the set of admissible input values.
Well-defined function
Every well-defined function must have a domain for which every admissible input argument has just one output value.
Examples
? is well-defined, because we have removed from the domain the one argument where the output is not defined. We have removed the singularity at .
? is not well-defined, because we have two possible output values for any given input argument.
? is well-defined. Since , every positive number has just one square root, which is also positive, so the output value of is determined for every input value of x.
SLIDE 18
In each of the following pairs of functions, one is well-defined, and the other is not well-defined. State which is which.
?
?
?
SLIDE 18B
In each of the following pairs of functions, one is well-defined, and the other is not well-defined. State which is which.
?
?
?
SLIDE 19
?
?
?
SLIDE 19B
?
?
?
SLIDE 20
Sets of numbers
The domain of a function is the set of numbers for which it is well-defined. A domain is a set of numbers.
? Revision
What is the difference between the following?
? A natural or counting number
? An integer
? A fraction or rational number
? A real number
SLIDE 21
? A natural or counting number is a number used to count things. It is the collection .
? An integer is a counting number, but allowed to be either positive or negative. It is the collection .
? A fraction or rational number is a ratio between two integers. For example, are rational numbers, because 2, 3, and 5 are integers. Fractions (rational numbers) can be both positive and negative.
? A real number is a collection of numbers that includes all the rational numbers, but also the irrational numbers, which are numbers like that cannot be written as rational numbers.
SLIDE 22
Domains of numbers
We introduce symbols for the sets of numbers
is the set of natural numbers
is the set of integers
is the set of rational numbers
is the set of real numbers, where is the set of irrational numbers
SLIDE 23
The domain of a well-defined function
Every well-defined function has a domain on which it is well-defined.
Therefore, strictly, we should always specify the domain when we define a function. Hence, every definition of a function should strictly consist of two parts: (1) a rule, and (2) a domain
Example
This definition consists two parts. (1) The rule for computing is given as , and (2) the domain is given as , meaning, any real number is a valid input argument for this function.
SLIDE 24
Implicit domain
If the domain is not given, or is given as an interval, then the implicit domain is the set of all real numbers, .
Note. is the interval that covers the whole of the line. We have, . It is the continuous interval.
Example
In this definition, the implicit domain is , saving that we have removed from the singularity at . The domain is the whole interval, less the one point .
SLIDE 25
General rule
The domain of every well-defined function is , unless something has been added explicitly to alter this.
Rationale. Functions arise in the context of measurement, and real numbers are used as measures of things. So we assume that the function is defined on the set of real numbers, unless otherwise told that it is not.
SLIDE 26
Restricting the domain
A not well-defined function may be converted to a well-defined function by removing from the domain , any numbers or interval where the definition fails to be well-defined.
This process is called restricting the domain.
Example
The domain has been restricted by removing from it the point . This converts the ill-defined rule into the well-defined function . In the definition, the domain is implicit.
SLIDE 27
In each of the following, explain how the domain has been restricted to produce a well-defined function.
?
?
?
SLIDE 27B
In each of the following, explain how the domain has been restricted to produce a well-defined function.
?
?
?
SLIDE 28
?
In the domain has been restricted by removing from the singularity at .
?
In the domain has been restricted to the interval , where x is a real number.
?
In the domain has been restricted to the interval , where x is a real number.
SLIDE 29
One-one or many-one functions
A function is one-one if for output value y there is only one input argument x such that .
A function is many-one if for some output value y there is more than one input argument x such that
Examples
? is one-one.
? is many-one. For example, there are two input arguments that give the same output value 4. These are and . In fact, only for is there only one input argument .
SLIDE 30
? Sketch separate diagrams of the graphs of the functions and
? Use these sketches to illustrate why is one-one, and is many-one.
SLIDE 31
The first sketch shows that the function is one-one because for every value of y there is only one argument x.
The second sketch shows that the function is many-one because for every value of there are two arguments of x that give y.
SLIDE 32
Increasing and decreasing functions
increasing decreasing
is an example of an always increasing function. The graph shows that the value of function is always going up for increasing x.
is an example of an always decreasing function. The graph shows that the value of the function is always going down for increasing x.
These expressions are abbreviated to increasing and decreasing respectively.
Advanced. A function is increasing if, and only if, implies . A function is decreasing if, and only if, implies .
SLIDE 33
A function is one-one if, and only if, it is either increasing or decreasing.
Is this statement true or false?
SLIDE 34
A function is one-one if, and only if, it is either increasing or decreasing.
This statement is true. If a function is not one-one then its graph must show at least one turning point – a maximum or minimum where the graph turns around and goes up, if it was originally going down, or down, if it was originally going up.
SLIDE 35
? ?
? ?
For each function above, state whether it is increasing, decreasing or many-one.
SLIDE 36
? ?
many-one increasing
? ?
decreasing many-one
SLIDE 37
The diagrams show the relationship between a function f and its inverse .
SLIDE 38
The figure shows a sketch of a cubic function .
By completing the above sketch explain why f cannot have an inverse.
SLIDE 39 – SOLUTION AND EXPLANATION
A function that is one-many does not have a well-defined inverse.
As the above figure illustrates, for the function there are many arguments of x that compute to the same values of the given y. We cannot define an inverse .
Important exception
If we restrict the domain of to any part where it is either always increasing or always decreasing, then on that part is one-one, and then we may define and inverse.
SLIDE 40
Switching variables
The figure shows the graph of the linear function .
? Find . By switching variables, write as a function of x.
? Draw onto the figure the graph of the function
SLIDE 41
SLIDE 42
? Compare the sizes of the two rectangles
? Describe a transformation that takes fits one rectangle onto the other.
? In terms of a find
SLIDE 43
? The two rectangles are the same size.
? Each rectangle is the reflection of the other in the line
?
SLIDE 44
Graph of an inverse function
Graph of ? reflect the graph of f in the line .
The graph of the inverse of a function f is the reflection in the line of the graph of .
SLIDE 45
? ?
The figures show the graphs of two linear functions. Make copies of the figures and add to each a sketch of the inverse function.
SLIDE 46
? ?
SLIDE 47
The figure shows the graph of .
? Explain why the restriction of the domain of f makes it possible for f to have an inverse.
? Give a well-defined definition of .
? By reflecting the graph of in the line add to the figure a sketch of the graph of .
SLIDE 48
is the restriction of to the positive half of the x-axis. By restricting in this way, the many-one function becomes the one-one, increasing function f. Hence f has an inverse.
The figure shows the graph of , called the square-root, as the reflection of f in the line .
SLIDE 49
The figure shows the graph of .
? Find the rule for .
? Give a well-defined definition of .
? By reflecting the graph of in the line sketch of the graph of . What do you observe?
SLIDE 50
.
? .
? is a well-defined definition of f.
? The graph of is the same as the graph of f. This is because . The inverse of f is the same as f.
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