How functions are presented to us
A taxi, a car park, a box made from cut-out cardboard: almost no real function is handed to you as a formula any more. You have to build it from a description, a table or a graph, and that is just as important as calculating with it.
Four forms of the same function
A car park charges $2$ € to enter and $1{,}50$ € per hour:
| Form | Example |
|---|---|
| Description | "2 € to enter and 1,50 €/hour" |
| Formula | $C(x) = 1{,}50x + 2$ |
| Table | hours: 0, 1, 2, 3 → euros: 2, 3,5, 5, 6,5 |
| Graph | points in a straight line, increasing |
Express through an algebraic formula the following relationship described verbally: "The cost $C$ of buying $x$ kilograms of apples if each kilo costs $1,80$ € and the…
Express through an algebraic formula the following relationship described verbally: "The cost $C$ of buying $x$ kilograms of apples if each kilo costs $1,80$ € and the plastic bag to carry them costs $0,10$ €."
A paid car park charges a base fee of $2$ € to enter and $1,50$ € for each hour of stay. Create a table of values showing the total cost for stays of $0$, $1$, $2$, $3$,…
A paid car park charges a base fee of $2$ € to enter and $1,50$ € for each hour of stay. Create a table of values showing the total cost for stays of $0$, $1$, $2$, $3$, and $4$ hours.
From a table or graph to a formula, and vice versa
If a table always increases by the same amount for each unit of $x$, the function is linear $y = mx + n$: $m$ is how much $y$ goes up each time, and $n$ is the value when $x = 0$.
To read a graph and obtain a table, choose several values of $x$ marked on the axis and write down the corresponding $y$.
Observe the following graphs and state which of them represent a mathematical function and which do not. Justify your answer using the vertical line test.
Observe the following graphs and state which of them represent a mathematical function and which do not. Justify your answer using the vertical line test.
Determine the algebraic expression of the linear function that relates the variables $x$ and $y$ given in the following table of values: $x$: $1$, $2$, $3$, $4$, $5$…
Determine the algebraic expression of the linear function that relates the variables $x$ and $y$ given in the following table of values:
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$x$: $1$, $2$, $3$, $4$, $5$
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$y$: $5$, $7$, $9$, $11$, $13$
From the following graph, construct a table of values for the coordinates $x \in \{-4, -2, 0, 2, 4\}$. You can do it on paper and send a photo with your answer.
From the following graph, construct a table of values for the coordinates $x \in \{-4, -2, 0, 2, 4\}$.
You can do it on paper and send a photo with your answer.
Observe the following tables of values and determine which one represents a function and which one does not. Justify your answer by stating the condition a relationship…
Observe the following tables of values and determine which one represents a function and which one does not. Justify your answer by stating the condition a relationship must satisfy to be a function.
Table A:
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$x$: $-2$, $0$, $2$, $2$, $4$
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$y$: $4$, $0$, $3$, $5$, $16$
Table B:
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$x$: $-2$, $-1$, $0$, $1$, $2$
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$y$: $4$, $1$, $0$, $1$, $4$
Piecewise-defined functions
A transport pass costs $20$ € per month with up to $30$ journeys included; each extra journey costs $0{,}50$ €:
$$C(x) = \begin{cases} 20 & \text{if } 0 \le x \le 30 \\ 20 + 0{,}50(x - 30) & \text{if } x > 30 \end{cases}$$Translate the following situation into a piecewise algebraic expression: "A youth travel pass costs $20$ € per month and allows up to $30$ journeys at no extra cost.…
Translate the following situation into a piecewise algebraic expression: "A youth travel pass costs $20$ € per month and allows up to $30$ journeys at no extra cost. From journey number $31$ onwards, $0,50$ € is charged for each extra journey made." Let $x$ be the total number of journeys made in the month and $C(x)$ be the total monthly cost.
Contextual domain: what makes sense in the problem
A $30$ cm piece of wire is bent to form a rectangle of base $x$: $A(x) = x(15-x)$. Mathematically $A$ exists for any $x$, but it only makes sense if $0 < x < 15$ (the base and the height must be positive).
A piece of wire $30$ cm long is available to form a rectangle. 1. Express the area of the rectangle $A(x)$ as a function of the measurement of its base $x$. 2. What is…
A piece of wire $30$ cm long is available to form a rectangle.
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Express the area of the rectangle $A(x)$ as a function of the measurement of its base $x$.
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What is the logical domain of this function in the context of the problem? That is, what real values can the base $x$ take for the rectangle to exist?
A square sheet of cardboard with side length $40$ cm is available. An open box is to be constructed by cutting a square of side $x$ from each of the four corners and…
A square sheet of cardboard with side length $40$ cm is available. An open box is to be constructed by cutting a square of side $x$ from each of the four corners and folding the resulting flaps upwards.
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Express the volume of the box, $V(x)$, as a function of the side $x$ of the cut-out square.
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Determine the domain of definition of this function in the geometric context of the problem.
In a physics experiment, a stone is dropped from a bridge $45$ metres above a river. The height $h$ of the stone above the water, measured in metres, as a function of…
In a physics experiment, a stone is dropped from a bridge $45$ metres above a river. The height $h$ of the stone above the water, measured in metres, as a function of time $t$ in seconds, is given by the expression $h(t) = 45 - 5t^2$.
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Construct a table of values for $t = 0$, $t = 1$, $t = 2$, and $t = 3$.
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At what exact moment in time does the stone hit the water?
You can do it on paper and send a photo with your answer.
Back to the beginning: The taxi mentioned at the start follows the same structure: if it charges a $3{,}15$ € minimum fare and $1{,}20$ € per kilometre travelled, its cost is $C(x) = 1{,}20x + 3{,}15$ with a contextual domain of $x \ge 0$.