Functions with roots

Determine the domain of definition of the following function with an even-index root:

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

Find the domain of the quadratic function under a square root:

$h(x) = \sqrt{16 - x^2}$

In a 5th Year exam, Hugo has to find the domain and the intercept points with the axes of the function $f(x) = \dfrac{5}{\sqrt{6 - 2x}}$. Hugo writes on his paper:

"Since there is a square root, the radicand cannot be negative: $6 - 2x \ge 0 \implies -2x \ge -6 \implies x \le 3$. Therefore, $\text{Dom}(f) = (-\infty, 3]$. Furthermore, to find the $X$-intercept, I set $6 - 2x = 0$, which yields $x = 3$; thus, the curve intersects the x-axis at $(3, 0)$".

1. a) Identify the two conceptual errors made by Hugo and explain with reasoning why they are incorrect according to the definition of functions with radicals in denominators.
2. b) Provide the corrected solution: determine the exact domain in interval form and prove whether or not the graph intersects each of the coordinate axes.

Calculate the domain of the function containing an even-index root in the denominator:

$f(x) = \frac{1}{\sqrt[4]{x^2 - 5x + 6}}$

Determine the values of the parameter $k$ for which the domain of the function $f(x) = \sqrt{x^2 + kx + 9}$ is the entire set of real numbers $\mathbb{R}$.

Calculate the domain of definition of the function with an odd-index root:

$g(x) = \sqrt[3]{2x^2 - 8x + 1}$

Find the domain of the irrational function containing a rational expression:

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

Find the domain of definition of the function formed by the sum of two roots:

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

Determine the domain of the irrational function whose radicand is an algebraic fraction with a quadratic numerator:

$f(x) = \sqrt{\frac{x^2 - 9}{x - 1}}$

In a technical study to mitigate the noise impact of a high-speed railway line in County Kildare, engineers model the acoustic attenuation $A(x)$ (in decibels, $\text{dB}$) provided by a green barrier using the function:

where $x$ represents the separation distance between the barrier and the centreline of the railway track (in metres).

1. a) Model the mathematical conditions required for the root to exist in $\mathbb{R}$: set up and solve the rational inequality that determines the domain of definition of $A(x)$. Explain in detail why the endpoint $x = 5$ must be excluded from the domain while the endpoint $x = 25$ is included.
2. b) Interpret the physical meaning of the endpoint value $x = 25\text{ m}$: what sound attenuation does the barrier provide at that distance according to the model?
3. c) A technician suggests installing the deflector barrier at $x = 30\text{ m}$ from the track to facilitate the passage of machinery. Use the theory of functions with radicals to justify why this proposal is unfeasible according to the adopted mathematical model.

Find the domain of definition of the nested function:

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

Determine the domain of the irrational function containing a rational expression where the numerator and denominator share roots:

$f(x) = \sqrt{\frac{x^3 - 3x^2 - 4x + 12}{x^2 + x - 2}}$

Consider the family of functions with a nested radical defined by:

where $k$ is a constant real parameter.

1. a) Formally prove that if $k < 0$, the domain of the function is the empty set ($\text{Dom}(f_k) = \emptyset$). Next, analyse what happens in the boundary case $k = 0$: is the domain still empty or is it formed by isolated points? Justify your answer by solving the corresponding inequalities.
2. b) For any fixed value $k > 0$, deduce the general expression for the domain of $f_k(x)$ in terms of $k$, expressing it as the union of two closed intervals symmetric with respect to the origin of coordinates.
3. c) (Inverse problem) Determine with reasoning the exact value of the parameter $k > 0$ required for the domain of the function to consist of two disjoint closed intervals whose combined total length (sum of the lengths of both intervals) is exactly equal to $6$ units.

Solve the inequality associated with the domain of the function $f(x) = \sqrt{\sqrt{x + 7} - x - 1}$ and express it in interval notation.

From the function $f(x) = \sqrt{x}$, write the algebraic expression for the function $g(x)$ whose graph is obtained by translating $f(x)$ three units to the left and two units upwards. What is the domain of $g(x)$?

Determine the domain and range of the function with a negative leading coefficient:

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

Observe the attached graph representing a function of the form $f(x) = a\sqrt{x - b} + c$. Determine the values of the parameters $a$, $b$ and $c$, knowing that the vertex of the semi-parabola is located at the point $(2, 1)$ and it passes through the point $(6, -3)$.

You want to design a function of the form $f(x) = a\sqrt{x - b} + c$ (with $a, b, c \in \mathbb{R}$) whose graph is a semi-parabola and which simultaneously satisfies the following three conditions:

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

2. Its range or image is $(-\infty, 5]$.

3. Its graph intersects the x-axis at the point $(13, 0)$.

1. a) Using exclusively the domain and range conditions, deduce the coordinates of the vertex $(b, c)$ of the semi-parabola and justify, without performing numerical calculations, what sign the coefficient $a$ must have.
2. b) Set up the corresponding equation with the intercept point $(13, 0)$ and calculate the exact value of parameter $a$.
3. c) Using the complete formula for $f(x)$, justify analytically whether the function is increasing or decreasing on its domain.

Find the axis intercepts of the function:

$f(x) = \sqrt{x + 9} - 3$

Calculate the points of intersection between the irrational function $f(x) = \sqrt{2x + 5}$ and the linear function $g(x) = x + 1$.

Given the function $f(x) = 3 + \sqrt{2x - 4}$, determine its inverse function $f^{-1}(x)$ and state the domain and range of this inverse function.

Given the function $f(x) = \sqrt{x^2 - 4x}$, calculate its inverse function $f^{-1}(x)$ by restricting the original domain to the interval $(-\infty, 0]$. Clearly state the domain and range of the resulting inverse function.

To solve the irrational equation $\sqrt{2x - 6} = 4 - \sqrt{x - 1}$, two 5th Year students preparing for the Leaving Certificate propose completely different approaches:

• Aoife's method (algebraic): Rewrite the equation as $\sqrt{2x - 6} + \sqrt{x - 1} = 4$, isolate one radical, square both sides twice consecutively, solve the resulting quadratic polynomial equation, and check the obtained solutions.

• Cian's method (analytical and functional): Study the joint domain of existence of both expressions, analyse the monotonicity (increase and decrease) of the functions $f(x) = \sqrt{2x - 6}$ and $g(x) = 4 - \sqrt{x - 1}$ to deduce how many real solutions there can be at most, and from there test integer values directly within the admissible interval.

1. a) Fully develop Aoife's method: show the step-by-step algebraic process, obtain the two solutions of the final quadratic equation, and explain why one of them is false (an extraneous solution) introduced by squaring both sides.
2. b) Develop Cian's method: determine the common domain, prove why $f$ is strictly increasing and $g$ is strictly decreasing on that domain, justify how many intersection points they can have at most, and find the exact solution.
3. c) Compare both methods: evaluate what algebraic risks Aoife's approach carries compared to Cian's functional approach, and justify in what type of problems involving radicals the functional method is more powerful.

The relationship between the area $A$ of a circle and its radius $r$ is given by the formula $A = \pi r^2$. Express the radius in terms of the area, $r(A)$, and state the valid domain of definition in this geometric context.

The period $T$ (in seconds) of a simple pendulum can be approximated by the function $T(L) = 2\pi\sqrt{\frac{L}{9.8}}$, where $L$ is the length of the pendulum in metres. Calculate the length required for the period to be exactly $2$ seconds and state the theoretical domain of the function $T(L)$ in this physical context.

The production cost $C(x)$, in euros, of $x$ parts of an electronic component for satellites follows a model of diminishing returns given by $C(x) = 2500 + 120\sqrt{x - 50}$. Determine the valid domain for the cost function and calculate how many parts were manufactured if the total cost amounted to $4900$ euros.

According to road safety expert reports from the Road Safety Authority (RSA), the estimated speed $v$ (in $\text{km/h}$) at which a vehicle was travelling before performing an emergency stop on a dry road can be approximated by the function $v(d) = 15.8 \times \sqrt{d}$, where $d$ is the length of the skid mark left on the asphalt (in metres).

In a police report for an incident on a road with a speed limit of $90\text{ km/h}$, the measured skid mark is exactly $d = 30\text{ m}$ long.

1. a) Before calculating the exact value with a calculator, estimate the speed with reasoning by bounding the value of $v(30)$ between two consecutive tens (hint: approximate $\sqrt{30}$ by comparing it to the closest integer perfect squares).
2. b) Now calculate the numerical value of $v(30)$ rounded to the nearest tenth, check it against your estimate, and determine whether the vehicle was exceeding the speed limit of the road.
3. c) If the skid mark had been twice as long ($d = 60\text{ m}$), does that mean the car was travelling at twice the speed? Reason your answer by analysing how the growth rate of the square root function changes as the independent variable increases.

In the theory of special relativity, the Lorentz factor $\gamma$ depends on the velocity $v$ of an object according to the function $\gamma(v) = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $c$ is the speed of light in a vacuum (approximately $3 \times 10^8$ m/s). Determine the domain of the function $\gamma(v)$ and interpret physically what happens to the function as $v$ approaches the boundaries of this domain.

A rectangle is inscribed in the region bounded by the x-axis and the upper semicircle of radius $R = 5$, whose equation is $y = \sqrt{25 - x^2}$. Express the perimetre of the rectangle $P(x)$ in terms of the coordinate $x$ of the vertex located in the first quadrant. Determine the valid domain of $P(x)$ in this geometric context and calculate its value for $x = 3$.

An engineering company estimates that the daily revenue $I(x)$ and costs $C(x)$, in thousands of euros, when manufacturing $x$ tonnes of a composite material are given by $I(x) = x\sqrt{40 - x}$ and $C(x) = 2x + 5$. Determine the domain of the profit function $B(x) = I(x) - C(x)$ in the context of the problem and calculate whether the company makes a profit or a loss when manufacturing $15$ tonnes.