Exercises: Domain, range, and image of a function

A student states that the domain of the rational function $f(x) = \frac{3x}{x^2+9}$ is $\mathbb{R} - \{-3, 3\}$. Explain the error in their reasoning when evaluating the denominator and write the correct domain of the function.

In an exam, a student states that the domain of the radical function $g(x) = \sqrt{12 - 3x}$ is the interval $[4, +\infty)$. Show mathematically why this result is a common mistake when solving inequalities and calculate the correct domain.

Determine the domain of definition of the following polynomial function:

$f(x) = -4x^5 + 7x^3 - x + 12$

Find the domain of the following rational function:

$f(x) = \frac{3x - 1}{2x - 8}$

Calculate the domain of definition of the radical function:

$f(x) = \sqrt{3x + 12}$

From the following graphical representation of the function $f(x)$, determine its domain and range.

Find the domain of the following rational function with a quadratic denominator:

$f(x) = \frac{x}{x^2 - 9}$

Determine the domain of the following logarithmic function:

$f(x) = \log_2(5 - x)$

A car park charges $2$ € for each hour or fraction of an hour that a vehicle remains parked, with a maximum limit of $12$ hours per day. Write the function relating the parking time $x$ (in hours) to the amount to be paid $f(x)$, and determine its domain and range in the context of this problem.

Calculate the domain of the following function containing a radical with an odd index:

$f(x) = \sqrt[3]{2x - 7}$

The public electric bicycle rental service in a city (such as BiciMAD in Madrid) charges occasional users a fee consisting of a fixed unlock fee of $0,50$ € plus $0,15$ € for each minute of the journey. According to road safety regulations and municipal bylaws, the maximum authorized time for a single uninterrupted journey is 2 hours ($120$ minutes). The function describing the final cost in euros based on the minutes used is $C(t) = 0,50 + 0,15 \cdot t$.

1. a) If you consider only the algebraic formula without taking into account the real-life situation, what is its mathematical domain and why?
2. b) Taking into account the regulatory restrictions of the service, determine the real domain and the range (or image) of the function, expressing both sets as intervals.
3. c) A user claims that the app charged them $22$ € for a single continuous journey compliant with regulations. Explain, relying on the concept of range, why that bill is not possible.

When solving an assignment about the domain of the rational function $f(x) = \frac{x + 5}{x - 3}$, Alejandro writes the following in his notebook:

"For the function to be computable, the numerator cannot be zero, so $x + 5 \ne 0$, which gives $x \ne -5$. Furthermore, the variable $x$ can never be a negative number, so the domain is the interval $[0, +\infty) \setminus \{5\}$".

1. a) Point out the two conceptual reasoning errors that Alejandro has made in his justification.
2. b) Explain why in a mathematical function the numerator of a fraction can indeed be zero and what numerical value $f(-5)$ takes exactly.
3. c) Correct the exercise by determining the correct domain of $f(x)$ and express it in two different ways: as an exclusion of values from $\mathbb{R}$ and as a union of intervals.

Determine the domain of the following function with a square root:

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

Find the domain of definition of the function that combines a fraction and a root:

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

Observe the graph of the piecewise function $g(x)$. Determine its domain and range, paying special attention to the vertical asymptote and to the points where the intervals are open or closed.

A rectangle is to be constructed using a piece of wire $40$ cm long. If $x$ is the length of the base of the rectangle, determine the expression for the area function $A(x)$ and find its domain of definition taking into account the geometric constraints of the problem.

A small toy rocket is launched vertically upwards from the ground. The height $h$ (in metres) it reaches as a function of time $t$ (in seconds) is given by the expression $h(t) = 20t - 5t^2$. Determine the domain and range of this function, knowing that the motion ends when the rocket hits the ground again.

Calculate the domain of the following logarithmic function with a quadratic argument:

$f(x) = \ln(9 - x^2)$

Determine the domain of the function that includes a root in the numerator and a polynomial in the denominator:

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

A company that manufactures electronic components has determined that the daily revenue from selling $x$ components is given by $I(x) = 150x - x^2$, and the total production costs by $C(x) = 2000 + 10x$. Knowing that the factory has a maximum production capacity of $100$ components per day, determine the profit function $B(x)$ and its valid domain in this economic context.

At an AEMET weather station located in the Sierra Nevada National Park, at an altitude of $2500$ m, the outdoor temperature $T$ (in $^{\circ}\text{C}$) recorded during a spring day fits the model $T(t) = -0,2 \cdot (t - 14)^2 + 8$, where $t$ represents the hour of the day measured continuously from midnight ($t \in [0, 24]$).

1. a) Before doing numerical calculations, make a reasoned estimate of what time of day you would expect to record the highest temperature and whether you consider it likely that in the early morning hours the temperature drops below $0\,^{\circ}\text{C}$.
2. b) Through algebraic analysis of the expression (or by calculating the vertex of the parabola), find the maximum temperature reached and evaluate the predicted temperatures at $00:00$ ($t = 0$) and at $24:00$ ($t = 24$).
3. c) Determine precisely the range (or image set) of the function for that entire day and check whether your initial estimate was consistent with the bounds obtained.

Imagine you are programming the safety parameters on a control screen for an industrial greenhouse:

1. a) Write the algebraic formula of a rational function $f(x) = \frac{1}{Q(x)}$ whose domain of definition is exactly $\mathbb{R} \setminus \{-2, 4\}$. Explain step by step how you construct the denominator $Q(x)$ to force the function to fail only at those two values.
2. b) Design the expression of a function with a quadratic radical $g(x) = \sqrt{P(x)}$ whose domain of existence starts at $6$ and extends to greater values, that is, $[6, +\infty)$. Justify your choice by solving the corresponding inequality.
3. c) Would it be possible to invent a polynomial function whose domain was exactly the interval $[6, +\infty)$? Justify your answer based on the properties of the domain of polynomials.

Determine the domain of the following function, which requires solving a rational inequality inside a square root:

$f(x) = \sqrt{\frac{x - 1}{x + 3}}$

Find the range of the following quadratic function analytically (remember to calculate the coordinates of its vertex):

$f(x) = -3x^2 + 12x - 5$

Calculate the values that parameter $k$ must take so that the domain of the following function is the entire set of real numbers $\mathbb{R}$:

$f(x) = \frac{2x - 7}{x^2 + kx + 16}$

From a rectangular sheet of cardboard of $40 \text{ cm} \times 25 \text{ cm}$, an open box is to be constructed. To do this, a square of side $x$ is cut out from each of the four corners and the edges are folded upwards. Determine the expression for the volume function $V(x)$ of the box and find its valid domain of definition.

Given the functions $f(x) = \sqrt{x}$ and $g(x) = x^2 - 5x + 4$, find the analytical expression for the composite function $(f \circ g)(x)$ and rigorously determine its domain.

Analyse the following graph of the function $h(x)$. Determine its domain, its range and the points where the function is not defined or has a discontinuity, stating what type it is.

Determine the domain of the following logarithmic function whose argument is a rational expression:

$f(x) = \log_2\left(\frac{x^2 - 4}{x}\right)$

Calculate the domain of the following rational function that includes a modulus in its denominator:

$f(x) = \frac{3x + 1}{|x - 3| - 5}$

The Ministry for the Ecological Transition and the Demographic Challenge (MITECO) records the water reserves of a reservoir in the southern Iberian Peninsula during a $90$-day drought period. Technicians estimate that the impounded volume $V(t)$, expressed in cubic hectometres ($\text{hm}^3$), follows the radical model $V(t) = \sqrt{3600 - 40 \cdot t}$, where $t$ represents the number of days elapsed since the beginning of summer ($t = 0$).

1. a) Determine the purely algebraic domain of the function $V(t)$ by solving the inequality imposed by the even-index root.
2. b) Explain the physical meaning of the value of $t$ that makes the radicand zero ($V(t) = 0$). Would it make practical sense to extend the domain beyond that point? Define what the contextualized domain of the model should be for monitoring during the $90$ days of the summer campaign.
3. c) Find the range of the function in the time interval of $90$ days ($t \in [0, 90]$) and interpret what information the extreme values of the range provide to the river basin managers.

To calculate the domain of the function $f(x) = \frac{1}{\sqrt{10 - 2x}}$, two students propose different approaches:

• Strategy 1 (Marcos): He applies the rules separately. First he solves $10 - 2x \ge 0$ because it is a square root, then he finds when the denominator is zero by solving $\sqrt{10 - 2x} = 0$, and finally he excludes that solution from the previous set.

• Strategy 2 (Sara): She groups both conditions into a single inequality. She argues that, since the radical is in the denominator, the radicand can be neither negative nor zero, so she directly sets $10 - 2x > 0$.

1. a) Work out both methods in detail and verify whether the identical domain is obtained with both.
2. b) Evaluate the two strategies: which do you consider more direct and with less risk of error when using parentheses or brackets to write intervals? Justify your choice.
3. c) If the function were $g(x) = \frac{\sqrt{10 - 2x}}{x - 3}$, reflect and explain whether Sara's simplified strategy would still work in a single step or whether it would be necessary to resort to Marcos's part-by-part analysis method.