Types of functions

A video game streaming platform offers a monthly subscription consisting of a fixed fee of $12$ € per month plus an additional cost of $0,50$ € for each hour of play in the "cloud".

1. a) Write the algebraic expression of the function $f(x)$ that relates the monthly cost to the number of hours played $x$.

1. b) State the slope ($m$) and the y-intercept ($n$) of this function.

1. c) How much will a student who has played $20$ hours in a month pay?

A municipal electric scooter rental service wants to model its costs. The total daily operating cost $C(x)$ (in euros) depends on the total kilometers traveled by the whole fleet $x$ and follows a linear function. It is known that on a day when the scooters traveled $20$ km the cost was €$15$, and on another day with $50$ km traveled the cost came to €$30$. The figure shows these two data points.

1. a) Determine the analytical expression of the function $C(x)$.
2. b) Identify and explain the meaning of the slope and the y-intercept in the context of the problem.
3. c) If the maximum daily budget is €$50$, how many kilometers can they travel at most?

Find the inverse function of the following linear function, which represents the conversion of in-game virtual currency into euros:

$f(x) = \dfrac{x - 100}{50}$

Follow the steps learned in class:

1. Substitute $f(x)$ with $y$.

2. Rearrange the variable $x$ in terms of $y$.

3. Swap the variables to obtain $f^{-1}(x)$.

Given the rational function $f(x) = \frac{2x - 5}{x - 3}$:

1. a) Find its inverse function $f^{-1}(x)$.

1. b) Prove analytically that the composition $(f \circ f^{-1})(x) = x$.

1. c) Graph both functions on the same Cartesian plane and draw the line of symmetry $y = x$. State the coordinates of the points where the function and its inverse coincide (invariant points).

The design of a new ramp for a skate park follows the shape of a square root function. For the structure to be stable, the engineer must calculate its domain.

Find the domain of the function:

$f(x) = \sqrt{2x - 8}$

Remember that for functions with even-index roots, the radicand must be greater than or equal to zero.

A forest surveillance drone analyzes the terrain following a trajectory whose height above the ground is given by the function $h(x) = \sqrt{2x + 8}$, where $x$ is the horizontal distance from the base of operations, in kilometers, and $h(x)$ is the height, in hectometers.

1. a) Determine the domain of the function in this context (take into account which values of $x$ make sense as a distance).

1. b) Calculate the average rate of change (ARC) of the height over the interval $[0, 4]$.

1. c) At what distance from the base does the drone reach a height of $4$ hectometers?

A young entrepreneur sells custom t-shirts online. He has decided to apply a bulk discount, defining the total price using the following piecewise function:

$f(x) = \begin{cases} 15x & \text{if } 0 \le x \le 5 \\ 12x + 15 & \text{if } x > 5 \end{cases}$

Where $x$ is the number of t-shirts purchased.

1. a) How much will it cost to buy $4$ t-shirts?
2. b) What is the price of an order of $10$ t-shirts?
3. c) Is the function continuous at the boundary point $x = 5$? Justify your answer by calculating the value of the function at that point for both sections.

A local council has established a new water tariff designed to encourage saving. The monthly cost $f(x)$ in euros for $x$ cubic metres consumed is defined by the following piecewise function:

$$f(x) = \begin{cases} 0,8x + 12 & \text{if } 0 \leq x \le 15 \\ 1,2x + k & \text{if } x > 15 \end{cases}$$

1. a) Find the value of $k$ so that the tariff is continuous at the transition point (so that there are no sudden jumps in the bill).
2. b) Sketch the graph of the resulting function.
3. c) How much will a family that consumes $22$ $m^3$ per month pay?

You can do it on paper and send a photo with your answer.

An online video game retailer offers a discount that depends on the total spend $x$ made. The function describing the final amount payable $G(x)$ after applying the discount is:

$$G(x) = \begin{cases} x & \text{if } 0 \leq x < 30 \\ 0,8x + 6 & \text{if } x \ge 30 \end{cases}$$

1. a) Study whether the function is continuous at $x = 30$. What does this result mean for the buyer?
2. b) Represent the function graphically.
3. c) If a user has paid $54$ €, what was the original price of their purchase before the discount?

You can do it on paper and send a photo with your answer.

Graph the absolute value function $f(x) = |x + 2|$.

To do so:

1. Identify the vertex of the function (the value of $x$ that makes the expression inside the absolute value equal to zero).

2. Create a table of values with two points to the left and two to the right of the vertex.

3. Draw the resulting graph on a Cartesian coordinate system, marking the vertex and the points from the table.

You can do it on paper and send a photo with your answer.

Plot the function $f(x) = |x^2 - 4x + 3|$. To do so, follow these steps:

1. a) Draw the parabola $y = x^2 - 4x + 3$ by finding its vertex and intercepts with the $OX$-axis.
2. b) Apply the definition of modulus to express $f(x)$ as a piecewise function.
3. c) State the intervals where the function is increasing and decreasing.

You can do it on paper and send a photo with your answer.

Find the analytic expression of the function $f(x) = |x + 2| + |x - 3|$. To do so:

1. a) Divide the real line into the intervals defined by the roots of the arguments of the absolute values.

1. b) Write the resulting function as a piecewise function, with the expression of $f(x)$ in each interval.

1. c) Graph it. What is this type of function, made up of straight-line segments, called? What is its absolute minimum value?

Given the inverse proportion function $f(x) = \dfrac{4}{x}$, perform the following tasks:

1. a) Complete the table of values for $x = \{-4, -2, -1, 1, 2, 4\}$.
2. b) State the asymptotes of the function.
3. c) Explain, giving reasons, why the value $x = 0$ does not belong to the domain of this function.

In a chemistry laboratory, the concentration of a reactant in a solution (in mg/L) as a function of time $t$ (in minutes) is given by the inverse proportion function:

$$C(t) = \dfrac{40}{t + 2}$$

1. a) What is the concentration at the initial time $t=0$?
2. b) Plot the function for values of $t \ge 0$.
3. c) Is it possible for the concentration to reach $0$ mg/L? Justify your answer by analysing the horizontal asymptote of the function.
4. d) At what time is the concentration exactly $5$ mg/L?

You can do it on paper and send a photo with your answer.