Exercises: Linear function; inverse; square root; piecewise; modulus

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?

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 $f^{-1}(x)$.

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.

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.

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.

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.

Given the functions $f(x) = 2x - 3$ and $g(x) = x^2 + 1$, calculate the following composite functions:

1. a) $(g \circ f)(x)$ [Remember that this is $g(f(x))$]
2. b) $(f \circ g)(x)$ [Remember that this is $f(g(x))$]
3. c) Evaluate the result of $(g \circ f)(2)$.

Observe the following graph showing the relationship between the temperature in a greenhouse and the time of day.

1. a) Determine the domain and range of the function based on the image.
2. b) State the intervals of increase and decrease.
3. c) At what time is the maximum temperature reached and what is this value?

Mateo has to calculate the inverse function of $f(x) = \dfrac{4x - 6}{5}$. On his exam he writes:

"Since the inverse function is the opposite operation, it is enough to invert the fraction: $f^{-1}(x) = \dfrac{5}{4x - 6}$. Furthermore, since the denominator cannot be zero, its domain is all real numbers except $x = 1,5$".

1. a) Explain with reasoning what conceptual confusion Mateo has between the concept of the reciprocal or multiplicative inverse of an expression, $\dfrac{1}{f(x)}$, and that of the inverse function with respect to composition, $f^{-1}(x)$.
2. b) Numerically show that Mateo's formula is incorrect by calculating the image $f(4)$ and verifying that applying his supposed inverse to the obtained result does not recover the initial $4$.
3. c) Find step by step the correct expression for $f^{-1}(x)$ by properly isolating the variable and state its domain.

The State Meteorological Agency (AEMET) uses a linear model in the troposphere to estimate the outside temperature $T$ (in $^\circ\text{C}$) as a function of altitude $h$ (in metres above sea level). On a winter day in the Sierra de Guadarrama, the temperature recorded at the plain level ($0\text{ m}$) is $11\ ^\circ\text{C}$ and drops at a rate of $0,65\ ^\circ\text{C}$ for every $100\text{ m}$ of vertical ascent.

1. a) Write the linear function $T(h)$, clearly identifying the slope $m$ (in $^\circ\text{C}/\text{m}$) and the $y$-intercept $n$.
2. b) Before performing the exact numerical calculation, estimate with reasoning whether at the summit of Peñalara ($2428\text{ m}$) the temperature will be above or below $0\ ^\circ\text{C}$ (mentally round the altitude to a nearby simple value).
3. c) Calculate the exact value of $T(2428)$ using the function and compare the result with your previous estimate.

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?

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.

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

1. a) Calculate its domain.
2. b) Find the axes intercepts.
3. c) Determine the inverse function $f^{-1}(x)$.
4. d) Check that $(f \circ f^{-1})(x) = x$.

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?

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.

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.

Find the domain of the following functions and classify their possible points of discontinuity:

1. a) $g(x) = \dfrac{x^2 - 1}{x^2 - 3x + 2}$

1. b) $h(x) = \sqrt{x^2 - 9}$

In each case, justify your answers: state what condition must be met for the function to be defined, factor when necessary and, for each point of discontinuity, study the corresponding limit to decide whether it is removable, a jump, or an (infinite) asymptote.

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.

Determine the algebraic expression of a radical function of the form $f(x) = a\sqrt{x - b} + c$ knowing that it simultaneously satisfies the following three conditions:

1. Its domain of definition is the interval $[4, \infty)$.

2. Its range or image set is the interval $[-1, \infty)$.

3. Its graph passes through the point $(13, 5)$.

With this information:

1. a) Deduce the values of the parameters $a$, $b$, and $c$, justifying how the radicand and the vertical shift influence the domain and range.
2. b) Find the inverse function $f^{-1}(x)$, detailing the algebraic process of isolating the variable.
3. c) Specify with reasoning the domain and range of $f^{-1}(x)$, relating them directly to those of the original function $f(x)$.

A public water supply company bills the bi-monthly consumption of a household using a progressive tiered tariff to penalise excessive consumption:

• Fixed service fee independent of consumption: €$8$.

• Tier 1 (up to $15\text{ m}^3$): each $\text{m}^3$ costs €$0,60$.

• Tier 2 (consumption over $15\text{ m}^3$ and up to $40\text{ m}^3$): the first $15\text{ m}^3$ are billed at the Tier 1 rate and each additional $\text{m}^3$ at €$1,40$.

• Tier 3 (consumption over $40\text{ m}^3$): the first $40\text{ m}^3$ are settled according to the previous tiers and each additional $\text{m}^3$ at €$2,80$.

1. a) Write the analytical expression of the continuous piecewise function $C(x)$ that gives the total bill amount in euros as a function of the volume $x$ of water consumed (in $\text{m}^3$).
2. b) Justify that the function does not have jump discontinuities by checking the values at the tier boundaries ($x = 15$ and $x = 40$).
3. c) If a family has received a bill for a total amount of €$80$, determine with reasoning which billing tier they are in and calculate exactly how many $\text{m}^3$ they consumed in that two-month period.

A 3D printing workshop has determined that the monthly production cost (in thousands of euros) is given by the function $C(x)$, where $x$ represents the number of batches produced. For logistical reasons, the cost structure changes when exceeding 10 batches:

$$C(x) = \begin{cases} x^2 + ax + 20 & \text{if } 0 \le x \le 10 \\ \sqrt{bx + 100} + 40 & \text{if } x > 10 \end{cases}$$

You are required to:

1. Determine the values of $a$ and $b$ so that the cost function is continuous throughout its domain, given that if nothing is produced the cost is €20,000, and if 10 batches are produced the cost is €120 000.

2. Having found $a$ and $b$, analyse whether there is any production level in the second section for which the cost stabilises at a fixed value.

Analytically study the domain of the following function, which combines radicals and algebraic fractions:

$$f(x) = \sqrt{\frac{x^2 - 5x + 6}{x + 4}}$$

To do this, you will need to solve the resulting rational inequality and express the final domain using interval notation. Justify whether the endpoints of the intervals should be included or excluded based on the definition of the function.

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).

An investment portfolio management application uses a function to model risk based on the exposure time $t$ (in months). The cumulative risk function is:

$$R(t) = |t^2 - 6t + 5|$$

You are required to:

1. Express the function $R(t)$ as a piecewise-defined function without using modulus bars.

2. Sketch its graph and identify the moments when the risk is zero.

3. Determine the time intervals, within the first 6 months, in which the risk is strictly increasing.

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?

In a market study, product prices were analysed at three different times:

• In month 1, the price was 10 €.

• In month 3, the price rose to 18 €.

• In month 5, the price stood at 22 €.

You are required to:

1. Use quadratic interpolation (Newton's method or a system of equations) to find the function $P(t)$ that best fits these data.

2. Estimate the expected price for month 7 using extrapolation.

3. Discuss whether this model is reasonable in the long term by analysing the vertex of the parabola obtained.

The supply ($O$) and demand ($D$) functions for a model of wireless headphones are given by the following expressions, where $p$ is the price in euros and $O(p)$, $D(p)$ are the quantities (in units):

$$O(p) = \frac{1}{2}p - 10$$ $$D(p) = \frac{800}{p}$$

1. a) Determine the market equilibrium point (price and number of units). Round to one decimal place.

1. b) If the government imposes a maximum price of 30 €, will there be excess supply or excess demand? Calculate that difference.

1. c) Graph both functions in the first quadrant, marking the equilibrium point and identifying the asymptotes of the demand function.

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

Given the function $f(x) = \sqrt{x + 1}$, you are required to:

1. Find its inverse function $f^{-1}(x)$, clearly specifying the domain and range of both.

2. Now consider a new function $g(x) = f(x - 3) + 2$. Explain in detail which geometric transformations (horizontal and vertical translations) have been applied to the original square root function to obtain $g(x)$.

3. Write the algebraic expression for $g(x)$ and calculate its y-intercept, if it exists.

Consider the function $f(x) = |x + 1| + |x - 5|$. To analyze it and find its global minimum value, two students propose different approaches:

• Method A (piecewise analytical, proposed by Irene): Divide the real line into three intervals based on the points where the absolute values become zero ($x = -1$ and $x = 5$). Expand the algebraic expression in each piece by removing the bars according to the sign and study the resulting slopes.

• Method B (geometric via distances, proposed by Tomás): Apply the definition of absolute value as the distance between two points on the real line ($|x - a| = d(x, a)$), interpreting the function as the sum of distances from a variable point $x$ to the fixed points $-1$ and $5$: $f(x) = d(x, -1) + d(x, 5)$.

1. a) Apply Method A: write $f(x)$ as a piecewise-defined function and determine its minimum value and on which set of values of $x$ it is achieved.
2. b) Apply Method B: using exclusively the visualization of distances on the real line, explain why the sum of distances cannot be less than the distance between $-1$ and $5$, and directly deduce the minimum value and the points where it is attained without developing algebraic pieces.
3. c) Compare both methods and justify which one is more suitable if you need to calculate the formal expression to differentiate or plot the entire graph, and which one is more efficient if you only need to quickly solve the inequality $|x + 1| + |x - 5| \le 10$.

Paula is trying to calculate the inverse function of $f(x) = 3 - \sqrt{2x - 4}$. Her procedure on the board is as follows:

1. She writes $y = 3 - \sqrt{2x - 4}$.

2. She isolates the square root: $3 - y = \sqrt{2x - 4}$.

3. She squares both sides: $(3 - y)^2 = 2x - 4$.

4. She isolates the variable $x$: $x = \dfrac{(3 - y)^2 + 4}{2} = \dfrac{y^2 - 6y + 13}{2}$.

5. She swaps the variables and concludes: "The inverse function is $f^{-1}(x) = \dfrac{x^2 - 6x + 13}{2}$, whose domain is all of $\mathbb{R}$ because it is a quadratic polynomial".

1. a) Point out the conceptual error made by Paula regarding the relationship between the range of $f$ and the domain of $f^{-1}$, and explain geometrically why a full parabola cannot be the inverse of a square root branch.
2. b) Show that the formula obtained by Paula fails for $x = 5$: calculate what value $f^{-1}(5)$ produces with her formula and check whether applying $f$ to that result recovers the original value $5$. Explain why the algebraic step of squaring both sides generated this problem.
3. c) Provide the correct solution: determine the exact domain and range of $f(x)$, and write the analytical expression of $f^{-1}(x)$, obligatorily indicating its restricted domain of definition.