The Concept of a Function

Observe the following relationships between quantities and determine, in each case, whether it is a function or not:

1. a) Each cell phone number is associated with the name of its owner in the contact list.
2. b) Each student in your class is associated with their height measured in centimeters.
3. c) Each supermarket barcode is associated with the price of the corresponding product.
4. d) Each person is associated with the eye color of their siblings.

At a secundaria (middle school), different relationships between quantities are being analyzed. Determine, giving reasons, which of the following situations correspond to a function and, if so, identify the independent variable and the dependent variable:

1. a) Each student's ID number is associated with their current age.
2. b) Each maximum temperature reached in the schoolyard during a day is associated with the times at which that temperature was recorded.
3. c) The amount of gigabytes ($GB$) consumed on a cell phone plan is associated with the monthly bill price.
4. d) Each natural number is associated with its factors.

In an app store, each premium game has a fixed price of $50$ pesos. We want to analyse the relationship between the number of games purchased and the total cost of the purchase.

1. a) Identify which is the independent variable and which is the dependent variable.
2. b) Is this relationship a function? Give reasons for your answer based on the definition of a function.
3. c) Write the algebraic expression (formula) that relates both quantities, letting $x$ be the number of games and $y$ be the total cost.

Given the function defined by the algebraic expression $f(x) = 3x - 2$, complete the following table of values in your notebook:

Once completed, state the image of $x = -1$.

The price of a monthly subscription to a streaming service is $180$ pesos, but there is a special offer for the first month and it only costs $60$ pesos.

1. a) Draw up a table relating the number of subscribed months (from $1$ to $4$ months) to the total cumulative expenditure.
2. b) Plot these four points on a Cartesian coordinate system.
3. c) Does it make sense to connect the points with a continuous line? Explain why.

Given the function $f(x) = \frac{x^2 - 4}{2}$, complete the following tasks:

1. a) Calculate the images of the values $x = 0$, $x = 2$ and $x = -4$.
2. b) Determine the value of $x$ (pre-image) whose image is $f(x) = 6$.
3. c) Does the point $P(-1; -1.5)$ belong to the graph of the function? Justify your answer by showing the necessary calculations.

For a graph to represent a function, any vertical line we draw must cut the graph at a single point. Use this "vertical line test" to decide which of the following graphs is NOT a function:

Justify your choice by explaining what happens on the chosen graph when you draw a vertical line.

Observe the following graphs plotted on the Cartesian plane. Using the vertical line test, state with reasons which of them represents a function and which does not.

A hiker climbs a mountain following a trail. The function that relates their altitude above sea level, $H$ (in meters), to the horizontal distance covered, $d$ (in kilometers), is $H(d) = 400 + 150d$.

1. a) At what altitude is the starting point of the route ($d = 0$)?
2. b) If the hiker has covered $3$ km horizontally, at what altitude are they?
3. c) Identify the variables of the function and explain their physical meaning.

A hiker records the percentage of battery remaining on their smartphone throughout a mountain walk. The resulting graph is as follows:

From the graph, answer the following:

1. a) Which are the independent and dependent variables? State the units of measurement.
2. b) What was the battery percentage at the start of the hike?
3. c) Is it an increasing or decreasing function? What is the physical meaning of this in the context of the problem?
4. d) Estimate how much time has passed when the phone has exactly $50$% charge remaining.