Exercises: Basic concepts

Given the polynomial function $f(x) = 2x^2 - 5x + 1$, calculate the images for the values $x = -2$, $x = 0$ and $x = 3$.

A bicycle hire service charges a fixed fee of $3$ euros to register for the service, plus $1,50$ euros per hour of use. Write the algebraic expression of the function that relates the total cost to the time of use, and identify which is the independent variable and which is the dependent variable.

Calculate the average rate of change of the function $f(x) = x^2 - 4$ over the intervals $[-1, 2]$ and $[2, 5]$.

Determine the domain of definition of the following functions:

1. a) $f(x) = \frac{3}{x - 5}$
2. b) $g(x) = \sqrt{x + 2}$

Observe the following graph of a function $f(x)$ and determine its intervals of increase and decrease, as well as the exact coordinates of its local maximums and minimums.

Determine algebraically whether the following functions are even, odd, or have no symmetry:

1. a) $f(x) = x^4 - 2x^2$
2. b) $g(x) = x^3 - 3x + 1$

From the graph of the function $h(x)$, knowing that the $y$-coordinate of the open point at $x = -2$ is $1,25$, state its domain and range in interval notation.

Calculate analytically the points of intersection with the x-axis and the y-axis of the function $f(x) = x^2 - 6x + 8$.

When studying the function $f(x) = \dfrac{2x - 6}{x + 1}$, Hugo makes a very common conceptual error when finding the intercepts with the axes. In his notebook, he writes:

"To find the intercept with the $Y$-axis, I solve the equation $2x - 6 = 0$, from which I get $x = 3$, so the graph intersects the $Y$-axis at the point $(3, 0)$. To find the intercept with the $X$-axis, I substitute $x = 0$ and calculate $f(0) = \dfrac{-6}{1} = -6$, so it intersects the $X$-axis at the point $(0, -6)$".

1. a) Explain in detail what confusion Hugo had regarding the roles of the independent and dependent variables when finding the intercepts with the axes.
2. b) Reason, from the definition of a function, why it is mathematically impossible for the graph of a function to intersect the $Y$-axis at two or more distinct points.
3. c) Correct Hugo's procedure and determine the correct coordinates of the intersection points of $f(x)$ with both Cartesian axes.

Design the algebraic expression of a rational function of the form $f(x) = \dfrac{a \cdot x + b}{x + c}$ that simultaneously satisfies the following three conditions:

1) Its domain of definition is $\mathbb{R} \setminus \{2\}$.

2) Its graph intersects the axis of ordinates ($Y$-axis) at the point $(0, -3)$.

3) Its graph intersects the axis of abscissas ($X$-axis) at the point $(3, 0)$.

Explain step by step how you deduce the numerical value of each of the parameters $a$, $b$, and $c$, and write the final formula of the function, verifying that the numerator does not vanish at the value excluded from the domain.

The distance $d$ (in metres) covered by a free-falling object as a function of time $t$ (in seconds) is approximated by the function $d(t) = 5t^2$. Calculate the average speed of the object (which corresponds to the average rate of change) between the first and third seconds ($t \in [1, 3]$), and between the third and fifth seconds ($t \in [3, 5]$).

Find the domain of definition of the function $f(x) = \frac{\sqrt{x + 4}}{x - 2}$.

Given the following graph of a piecewise-defined function, find the values of $f(-2)$, $f(1)$ and $f(4)$. Then, state at which points the function is discontinuous and the type of each discontinuity.

Determine analytically the equations of the vertical and horizontal asymptotes of the rational function $f(x) = \frac{2x + 1}{x - 3}$.

A taxi company charges $3$ euros standing charge and $1,20$ euros per kilometre travelled.

1. a) Write the function relating the total price of the journey to the kilometres travelled.
2. b) What is the domain of this function taking into account the context of the problem?
3. c) How much will a journey of $15$ km cost?
4. d) If a customer paid $21$ euros, how many kilometres did they travel?

The accompanying graph represents the height in metres of a Ferris wheel capsule as a function of time in minutes. Observe the graph and answer:

1. a) Is it a periodic function? If so, state its period.
2. b) What are the maximum and minimum heights reached by the capsule?
3. c) At what height is the capsule exactly $10$ minutes after starting to rotate?

Calculate the average rate of change of the function $f(x) = \frac{1}{x}$ over the intervals $[1, 2]$ and $[2, 4]$. What does the sign of the results indicate about the monotonicity of the function over those intervals?

The temperature $T$ (in degrees Celsius) inside an industrial oven from when it is turned on until it is turned off is given by the function $T(t) = -t^2 + 8t + 20$, where $t$ is the elapsed time in hours. At what moment is the maximum temperature reached? What is the value of this maximum temperature?

The temperature $T$ (in degrees Celsius) recorded by an AEMET meteorological station in a mountain area during a winter afternoon is given by the function $T(t) = -0,5 \cdot t^2 + 2t + 4$, where $t$ is the time elapsed in hours since 12:00 noon ($t = 0$), for $t \in [0, 6]$.

1. a) Without performing exact numerical calculations, qualitatively estimate whether the Average Rate of Change on the interval $[1, 5]$ (that is, $\text{ARC}[1, 5]$) will be positive or negative. Justify your estimate based on the shape of the parabola and the sign of its leading coefficient.
2. b) Algebraically calculate the exact value of $\text{ARC}[1, 5]$ applying its formula and state the physical units that correspond to the result.
3. c) Check whether the value obtained matches your previous estimate and explain what physical meaning that result has regarding the average cooling or warming of the environment during those hours.

The municipal electric bicycle rental service of a Spanish city bills the cost $C(t)$ (in euros) based on the usage time $t$ (in minutes, with $0 \le t \le 90$) using a piecewise function. The first 20 minutes combine an unlocking fee with a reduced rate, and after minute 20 the cost per minute increases to prevent monopolising the bikes:

$C(t) = \begin{cases} 0,60 + 0,05 \cdot t & \text{if } 0 \le t \le 20 \\ k + 0,15 \cdot (t - 20) & \text{if } 20 < t \le 90 \end{cases}$

1. a) Calculate the value that the parameter $k$ must take for the function to be continuous at $t = 20$. Reason what economic consequence or unfairness for the user would occur if there were a jump discontinuity at that moment.
2. b) Knowing that the tariff is continuous, if a user receives an exact charge of $4,60$ € on their mobile app, set up and solve the corresponding equation to find out how many minutes their journey lasted.
3. c) Calculate the Average Rate of Change of the cost on the interval $[5, 15]$ and on the interval $[30, 50]$. Explain what these two numerical values represent with respect to the price per minute of each tariff section.

Find the values of the parameters $a$ and $b$ so that the following piecewise-defined function is continuous throughout its domain:

$f(x) = \begin{cases} 2x + a & \text{if } x < 1 \\ x^2 - 3x + 5 & \text{if } 1 \le x < 4 \\ bx - 1 & \text{if } x \ge 4 \end{cases}$

A farmer has $100$ metres of fencing to enclose a rectangular vegetable patch using the straight wall of an existing building, so that he only needs to fence three of the four sides of the rectangle.

1. a) Find the expression of the function that gives the area of the patch as a function of the length $x$ of one of the sides perpendicular to the wall.
2. b) Determine the valid domain of definition for $x$ in the context of this problem.
3. c) What dimensions should the patch have so that its area is maximum?

Determine analytically the domain of definition of the function $f(x) = \sqrt{\frac{x - 3}{x + 2}}$. Express the result as a union of intervals.

Given the quadratic function $f(x) = x^2 + kx$, determine the value of the parameter $k$ given that the average rate of change of the function over the interval $[1, 3]$ is equal to $8$.

Analyse the following graph of the function $f(x)$ and determine:

1. a) Its domain and range.
2. b) The equations of its asymptotes.
3. c) The intervals of increase and decrease.
4. d) The coordinates of its local extrema (maximums and minimums).

Given the rational function $f(x) = \frac{x^2}{x^2 - 4}$:

1. a) Prove analytically whether it is an even function, an odd function, or has no symmetry.
2. b) Calculate the points of intersection with the coordinate axes.
3. c) Find the equations of all its asymptotes (vertical and horizontal).

A public car park has the following charges: it charges $2$ euros for the first hour (or fraction of an hour), and $1,50$ euros for each additional hour (or fraction). The maximum cost per day is $12$ euros. Write the analytical expression of the piecewise-defined function relating the price to be paid to the parking time $t$ (in hours) for the first $5$ hours, clearly indicating the intervals of definition for each section.

Given the rational function $f(x) = \frac{ax^2 + 3}{2x^2 - x}$, determine the value of the parameter $a$ so that the function has a horizontal asymptote at the line $y = 4$. With this value of $a$ calculated, also find the equations of the vertical asymptotes of the function.

To determine whether the rational function $f(x) = \dfrac{4x}{x^2 - 9}$ exhibits any type of symmetry (even, odd, or neither), two 4th-year ESO students defend different methods:

• Álvaro's strategy: He evaluates the function at a specific value in its domain, for example $x = 1$, obtaining $f(1) = -0,5$ and $f(-1) = 0,5$. Observing that $f(-1) = -f(1)$, he claims that this proves with certainty that the function is odd across its entire domain.

• Marta's strategy: She maintains that checking a particular case does not prove a general property. She proposes substituting the variable $x$ with $-x$ generically, simplifying the algebraic expression of $f(-x)$, and formally comparing it with $-f(x)$ and with $f(x)$.

1. a) Analyse Álvaro's strategy: reason why a numerical example does not constitute a valid mathematical proof and provide the formula of a function $g(x)$ that satisfies $g(-1) = -g(1)$ but is not odd on the rest of its domain.
2. b) Apply Marta's strategy by algebraically expanding $f(-x)$ and conclude with rigour what type of symmetry $f(x)$ has.
3. c) Justify in what exam situation Álvaro's method may be useful (as a way of ruling out or conjecturing) and why Marta's method is the only mathematically conclusive one.

An electronic engineering team designs the response of a signal filter whose voltage gain $f(x)$ (in volts) depends on the input frequency $x$ (in kHz). The response follows the rational function model:

$f(x) = \dfrac{a \cdot x^2 + b}{c \cdot x^2 - 18}$

For the circuit to operate optimally, it must strictly satisfy four technical specifications:

1) The graph is symmetric with respect to the $Y$-axis (even function).

2) The circuit enters critical resonance (exhibits vertical asymptotes) at exactly the frequencies $x = 3$ kHz and $x = -3$ kHz.

3) At extremely high frequencies ($x \to \infty$), the gain stabilises at a horizontal asymptote with a value of $y = 2$ V.

4) When there is no input signal ($x = 0$ kHz), the initial gain is $-1$ V.

1. a) Deduce, in a reasoned manner, the values of the coefficients $a$, $b$, and $c$. Justify each deduction from the given conditions.
2. b) Formally prove that the function obtained with these coefficients satisfies the algebraic condition of an even function for all $x \in \mathbb{R} \setminus \{-3, 3\}$.
3. c) Determine whether the output signal can vanish at any point (that is, whether the graph intersects the $X$-axis). Justify your answer by solving the equation $f(x) = 0$ and explain why having a horizontal asymptote at $y = 2$ does not, in principle, prevent a function from intersecting the horizontal axis.