Exercises: Properties of functions

Determine the domain of the rational function $f(x) = \frac{x-1}{x^2-9}$.

Find the domain of the radical function $g(x) = \sqrt{2x-8}$.

Find the axis intercepts of the polynomial function $h(x) = x^2 - 5x + 6$.

Study the symmetry (even, odd or neither) of the polynomial function $p(x) = x^4 - 2x^2 + 7$.

Determine the intervals in which the function $q(x) = -3x + 12$ is positive and those in which it is negative.

State the range of the quadratic function $r(x) = x^2 + 4$.

From the graphical representation of the parabola $y = f(x)$, state its intervals of increase and decrease, as well as the coordinates of its relative extremum.

Find the domain of the logarithmic function $s(x) = \log_3(x+2)$.

Carla states on her exam that the domain of definition of the function $f(x) = \ln(9 - x^2)$ is the closed interval $[-3, 3]$. To justify it, she writes in her working: “Since the domain needs to be calculated, I set up the inequality $9 - x^2 \ge 0$, whose solution is $x^2 \le 9$, so $x \in [-3, 3]$. I use square brackets because at $x = 3$ and at $x = -3$ the radicand equals zero and the square root of zero does exist”.

1. a) Find and explain in detail the two conceptual errors Carla made when identifying and applying the restrictions of elementary operations.
2. b) Reason what happens mathematically if we try to evaluate $f(3)$ on a scientific calculator and why the result cannot be a real number.
3. c) Correct Carla's procedure and write the correct domain of $f(x)$ using appropriate interval notation.

Design the analytic expression of a rational function $f(x) = \frac{P(x)}{Q(x)}$ (where $P(x)$ and $Q(x)$ are polynomials) that simultaneously satisfies the following three conditions:

1. Its exact domain is $\text{Dom}(f) = \mathbb{R} \setminus \{-2, 2\}$.

2. It is an even function, that is, symmetric with respect to the y-axis ($Y$).

3. It takes only strictly negative values ($f(x) < 0$) for any value of $x$ belonging to its domain.

Justify step by step how you choose the numerator and the denominator to guarantee each condition, and explain why a simple denominator of the form $Q(x) = x^2 - 4$ would not work if the numerator were a negative constant.

Find the domain of the function $f(x) = \sqrt{\frac{x+1}{x-3}}$.

Find the axis intercepts and study the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.

Determine analytically whether the function $h(x) = \frac{x^3}{x^2+1}$ is even, odd or has no symmetry.

A projectile is launched vertically upwards. The height $h(t)$ in metres that it reaches after $t$ seconds is given by the function $h(t) = -5t^2 + 20t + 1$. Determine at what time it reaches its maximum height and what this maximum height is.

A farmer has $100$ metres of fencing to enclose a rectangular vegetable garden taking advantage of an existing stone wall (no fencing is needed on that side). If $x$ is the length of the sides perpendicular to the wall, the function describing the area is $A(x) = x(100-2x)$. Determine the domain of the function in the context of the problem and calculate the dimensions that maximise the area.

Observe the graph of the following periodic function modelling the voltage in an electronic circuit. Determine its period $T$ and calculate the value of $f(17)$.

Given the piecewise function:

$f(x) = \begin{cases} 2x+6 & \text{if } x < -1 \\ x^2+3 & \text{if } x \ge -1 \end{cases}$

Find its axis intercepts and study its sign across its whole domain.

Determine the domain of the function $f(x) = \ln(x^2 - 4x + 3)$.

To study the sign of the function $f(x) = \frac{x^2 - 4}{x^2 + 1}$ across its entire domain, two classmates discuss which approach to follow:

• Strategy A (Traditional sign chart): Set numerator and denominator equal to zero separately, place all obtained solutions on the real line, and evaluate a test point in each resulting interval.

• Strategy B (Prior structural analysis): Analyze the overall sign of each term by algebraic inspection of its properties before breaking it down into intervals.

1. a) Apply Strategy B to determine, with reasoning, the intervals of positivity ($f(x) > 0$), negativity ($f(x) < 0$), and the x-intercepts.
2. b) Explain what algebraic difficulty or waste of time arises when trying to apply the denominator step in Strategy A over the set of real numbers ($\mathbb{R}$).
3. c) Justify which of the two methods is safer and more efficient to avoid mistakes when irreducible quadratic expressions appear in denominators.

In an AEMET report on winter thermal inversion in a mountain valley in Asturias, the ambient temperature $T$ (in $^{\circ}\text{C}$) is modeled as a function of altitude $h$ (measured in hundreds of meters above sea level, where $h = 5$ represents $500\text{ m}$) by the function:

1. a) Determine the mathematical domain of the function $T(h)$ from the restriction of the radical and compare it with the physical domain of the phenomenon if the valley floor is at $200\text{ m}$ ($h=2$) and the summit at $1000\text{ m}$ ($h=10$).
2. b) Without using derivatives, deduce the range (or image) of the function in that valley by reasoning about the maximum and minimum values the radicand can take. What are the predicted minimum and maximum temperatures?
3. c) At what altitude is the maximum temperature reached? Justify whether the function exhibits symmetry with respect to the vertical line $h = 6$ and interpret its physical meaning in the thermal distribution of the two slopes of the valley.

Find the domain of the function $f(x) = \sqrt{\log_2(x-3)}$, justifying your answer.

Determine the coefficients $a$, $b$ and $c$ of the quadratic function $f(x) = ax^2 + bx + c$ knowing that its vertex is the point $V(3, -4)$ and that it intersects the x-axis at $x = 1$.

The average monthly cost (in euros) per unit produced in an electronic components factory is given by the function $C_m(x) = \frac{300x + 1200}{x}$, where $x$ is the number of units produced ($x > 0$). Determine the domain in the context of the problem, study its sign and prove analytically in which intervals the average cost is increasing or decreasing.

Prove analytically that the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$ is odd and determine its domain.

A Ferris wheel has a radius of $10\text{ m}$, its center is located $15\text{ m}$ above the ground, and it rotates at a constant speed, taking $4\text{ min}$ to complete a full turn. A cabin is at its lowest point at time $t = 0\text{ min}$. Knowing that the height $h(t)$ in meters of the cabin as a function of time $t$ in minutes is modeled by a trigonometric function, determine its period, the maximum and minimum heights that the cabin reaches, and the times at which they are reached during the first complete cycle.

Find the values of the parameters $a$ and $b$ so that the function $f(x) = \frac{x^2+a}{x-b}$ has domain $\mathbb{R} \setminus \{2\}$ and intersects the x-axis at the points $x = 4$ and $x = -4$.

Find the inverse function of $f(x) = \frac{2x-5}{x+1}$ and use it to determine the range of the function $f(x)$.

From the graph of the function $y = f(x)$, perform a complete analysis stating: domain, range, symmetry, equations of the asymptotes, relative extrema and intervals of positive and negative sign.

Given the function with multiple operational restrictions $f(x) = \frac{\sqrt{9 - x^2}}{\ln(x)}$:

1. a) Estimate before operating:

• Without solving inequalities, reflect: does it make sense to consider whether this function is even or odd? Support your answer by analyzing negative values of $x$ against the components of the formula.

• Predict the sign that $f(x)$ will take for values very close to $0$ from the right (such as $x = 0{,}1$) and for values such as $x = 2$. Justify your prediction by recalling the elementary properties of logarithms.

1. b) Calculate and formalize:

• Determine rigorously the domain of definition of $f(x)$, explicitly setting up the three necessary conditions (fraction, even-index radical, and logarithm) and solving their intersection.

• Carry out the formal sign study of $f(x)$, indicating the exact intervals where $f(x) > 0$ and where $f(x) < 0$.

1. c) Assess and contrast:

• Check whether your initial estimates in (a) match the analytical calculation in (b).

• Reason whether the function has roots (points where $f(x) = 0$) and describe what behavior the function exhibits when $x$ approaches $1$ from the left and from the right.

In a bioengineering project, it is required to model the response $R(x)$ of a biosensor as a function of a control parameter $x \in \mathbb{R}$. You must construct the analytical formula of a function $f(x)$ that simultaneously satisfies the following four conditions:

1. Its exact domain is $\text{Dom}(f) = [-5, 5] \setminus \{0\}$.

2. It is strictly odd throughout its domain: $f(-x) = -f(x)$.

3. It has no roots in its domain (that is, $f(x) \ne 0$ for all $x \in \text{Dom}(f)$).

4. It takes strictly positive values when $x \in (0, 5]$.

1. a) Propose an algebraic expression for $f(x)$ combining elementary operations (algebraic fractions and even-index radicals). Justify how each element of your formula ensures the required conditions without canceling out the others.
2. b) Prove analytically that the function you designed satisfies the odd symmetry condition $f(-x) = -f(x)$.
3. c) Reasonably deduce the range (image) of your function on the interval $(0, 5]$ and, using odd symmetry, determine the complete range across its entire domain. Justify whether the function is bounded above by analyzing what happens when $x$ approaches $0$ from the right.