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Discussion
Suppose your phone charges from 0% to 100% in I4 hours, where I is the electric current (amps/hour) supplied by your charger. Imagine you can make the current as weak or strong as you want.
What are the possible values for I?
How long could your phone take to charge?
Part (a)
The charging time is given by
A zero or negative current would never charge the phone (zero current gives infinite time; negative would drain its battery). Therefore,
Part (b)
The charging time is
with I>0.
• Since I must be positive, time is always positive—there is no zero or negative charging time (it would be unrealistic for the phone to charge “instantly” or backwards in time).
• As I→+∞,
so the time can get arbitrarily small but never reach 0.
• As I→0+,
so the time can grow without bound if the current is made very small.
Therefore the charging time t can be any positive real number:
If we let t be the time, in hours, needed to charge the phone from 0% to full, then we have an equation t=I4. We say this equation defines a relation between time and current.
A relation between variables x and y is a set of points (x,y). Often, rather than listing points, relations are defined by rules that tell you the value of one variable in terms of another variable, like our example t=I4. This relation includes (I,t) points like (1,4), (2,2), and (3,34).
Theoretically, this relation also includes the point (−1,−4) even though we know this point does not make sense in context. Thus, you might define the relation as t=I4, where I≥0.
If a relation assigns exactly one output for each input value, then that relation is called a function. The time-current relation example is a function because, given any positive value for I, you could find the value of t. A non-example of a function would be y=±x because, given x=1, you could return either y=−1 or y=1.
Though the time to charge-current function is intuitive, many functions are harder to make sense of, which is why we study function theory - or the common properties and characteristics of functions.
Function vs relation
A function from x to y is a special type of relation where each x value has only one possible y-value. It is expressed in the form
where f and x can be replaced by any letters.
Discussion
We can rewrite the previous time-current relation as a function t(I)=I4. Find how many hours it will take to charge the phone if the current is 8 amps/hour.
We have the relation
and here I=8. Hence
We can write this as
Evaluating functions
A function can be evaluated for specific values of x by plugging the value into the expression of the function.
Example
Let f(x)=3−x2. Find f(3).
Exercise
Consider the function g(x)=7x+3.
Given that g(b−5)=17, find b.
Select the correct option
Recall our original questions: in what possible times can the phone charge, and what possible currents could be supplied by the charger?
The values you found were the domain and range of the function t.
Range of a function
The range of a function is the set of possible values it can output.
If the domain of the function is restricted, the range may need to be restricted as a consequence.
For example, if t(I)=I4 for I>0, then the range of t becomes 0<t(I)<∞. If, instead, the domain included negative I, then the range would become all real numbers except for 0.
Checkpoint
Find the range of f(x)=x2.
Select the correct option
Domain of a function
The domain of a function is the set of possible inputs it can be given.
The "natural" or "largest possible" domain of a function is all the values of x for which the expression f(x) is defined.
Checkpoint
Find the largest possible domain of f(x)=√x.
Select the correct option
Notation for domain and range intervals
The domain and range of functions are commonly intervals of real numbers.
For example, if f(x) is defined for 1<x≤5, we can write the domain
(∈ means "in" or "element of", and R is all real numbers)
We can also use the equivalent interval notation
where, by IB convention, an outward facing [ means that end is not inclusive (1<x) and an inward facing ] means that end is inclusive (x≤5).
Another common interval notation is
where ( indicates a non-inclusive endpoint and ] indicates an inclusive endpoint. In this style, all brackets are inward facing.
These can also be visualized on a number line:
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Note, some intervals can only be expressed in pieces. For example, −10≤x≤−2 or 4≤x≤7 cannot be written as one interval. In such cases, we use the union ( ∪ ) to adjoin two or more intervals:
Checkpoint
What is the range, in interval form, of f(x)=4x on the domain x∈[−2,8)?
Select the correct option
Exercise
For the function f(x)=√18−2x2 find
the domain,
the range.
Select the correct option
In the phone charging example, the function t(I)=I4 models a real-world situation in which the time to charge depends on the electric current.
Function as a model
A function as a model means using a mathematical relationship to represent real-world phenomena. By assigning input values (independent variables) and calculating corresponding outputs (dependent variables), a function allows us to approximate, describe, or predict patterns, behaviors, or outcomes.
For example, we can model the height H, in kilometers, of a rocket t minutes after its launch using a quadratic function:
0 / 5
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