The concept of a function
The battery level of your phone depends on the number of minutes you have been playing. The water in a tank depends on how long the tap has been open. The temperature during a day depends on the time. When one quantity depends on another in such a way that each value gives exactly one result, we have a function.
What a function is
One of the golden rules for deciding whether a graph represents a function is to check that for each value of the independent variable $x$ there is exactly one value of…
One of the golden rules for deciding whether a graph represents a function is to check that for each value of the independent variable $x$ there is exactly one value of the dependent variable $y$. State which of the following graphs is NOT a function and briefly justify your answer.
Independent and dependent variables
First of all, you need to identify what depends on what. For a scooter rental that charges €1.50 to unlock and €0.20 per minute:
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The time of use is chosen by the customer $\to$ independent variable ($x$).
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The total cost is calculated from the time $\to$ dependent variable ($y$).
To hire an electric scooter, a company charges $1,50$ € as an unlock fee and $0,20$ € per minute of use. We want to analyse the relation between the time of use and the…
To hire an electric scooter, a company charges $1,50$ € as an unlock fee and $0,20$ € per minute of use. We want to analyse the relation between the time of use and the total hire cost.
Identify the variables of this relation:
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Independent variable ($x$): ________
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Dependent variable ($y$): ________
An electric scooter has a battery that discharges at a constant rate. When leaving home, the battery is at $100\%$. For every kilometre travelled, the battery decreases…
An electric scooter has a battery that discharges at a constant rate. When leaving home, the battery is at $100\%$. For every kilometre travelled, the battery decreases by $5\%$.
- a) Identify the independent variable and the dependent variable, stating the units in which each is measured.
- b) Write the algebraic expression (formula) of the function relating the kilometres travelled ($x$) to the remaining battery percentage ($y$).
- c) Is it possible for the value of $x$ to be $25$? Reason whether that value belongs to the domain of the function in this real-world context.
The vertical line test
Not every curve drawn on the plane is a function. To check: draw imaginary vertical lines. If any of them cuts the graph at more than one point, it is not a function (that $x$ would have two different values of $y$).
Observe carefully the following three graphs drawn on the Cartesian plane. Use the vertical line test (imagine vertical lines sweeping across each graph and count at how…
Observe carefully the following three graphs drawn on the Cartesian plane. Use the vertical line test (imagine vertical lines sweeping across each graph and count at how many points they cross it) to identify which of them represent a function, and justify your answer in detail in each case.
Analyse carefully the following graphical representation showing the relationship between two quantities. Determine, reasoning your answer based on the definition of a…
Analyse carefully the following graphical representation showing the relationship between two quantities. Determine, reasoning your answer based on the definition of a function, whether this graph represents a function or a non-functional relationship. Also state what the value (or values) of the dependent variable $y$ would be when the independent variable $x$ is equal to $2$.
Reading a graph: domain, range and continuity
When reading the graph of a real function, always ask yourself: what is on each axis, in what units, and for which values does it make sense?
In a 20 L tank that fills at 4 L/min, the real domain is $[0, 5]$ minutes: after 5 min the tank is already full and the tap is turned off, so it makes no sense to continue the graph.
A graph can also have increasing sections (going up), decreasing sections (going down) and constant sections (flat), and it can be continuous (with no jumps or gaps) or have a jump somewhere.
The following graph shows the distance of a delivery drone from its logistics base during a 10-minute round trip. From the graph, state: a) The domain and the range of…
The following graph shows the distance of a delivery drone from its logistics base during a 10-minute round trip.
From the graph, state:
- a) The domain and the range of the function.
- b) How long was the drone stopped delivering the package? At what distance from the base was it at that time?
- c) In which time interval was the drone returning to the base?
We are filling a cylindrical water tank using a tap that pours a constant flow of $4$ liters per minute. At the start, the tank is empty and has a total capacity of $20$…
We are filling a cylindrical water tank using a tap that pours a constant flow of $4$ liters per minute. At the start, the tank is empty and has a total capacity of $20$ liters.
- a) State whether the function that relates the elapsed time to the volume of water is continuous or discrete. Explain why.
- b) What is the domain of this function, taking into account that the tap is turned off when the tank is full?
- c) Draw a sketch of the graph that represents this situation, with time $t$ (in minutes) on the horizontal axis and volume $V$ (in liters) on the vertical axis.
You can do it on paper and send a photo with your answer.