Exercises: Radical function

Determine the domain of definition of the function $f(x) = \sqrt{4x - 12}$.

Find the axis intercepts of the function $g(x) = \sqrt{3x + 9} - 3$.

Given the function $h(x) = \sqrt{5 - 2x}$, calculate the images corresponding to the values $x = -2$, $x = \frac{1}{2}$ and $x = -10$.

Look at the following graphs and match each of them (A, B, C and D) with its corresponding algebraic expression from the following options: $y = \sqrt{x+2}$, $y = \sqrt{x} + 2$, $y = \sqrt{x-2}$, and $y = \sqrt{x} - 2$.

Write the algebraic expression of the function resulting from translating the graph of the function $f(x) = \sqrt{x}$ three units to the left and five units down.

Find the value of $x$ for which the radical function $f(x) = \sqrt{15 - 3x}$ takes the value $y = 6$.

Give reasons to explain the difference between the domains of the functions $f(x) = \sqrt[3]{2x - 8}$ and $g(x) = \sqrt{2x - 8}$.

State the range of the functions $f(x) = \sqrt{x - 5}$ and $g(x) = -\sqrt{x - 5}$.

Sofía is studying the graph of the radical function $f(x) = \sqrt{x - 4} + 3$ and wants to find at which point it intersects the vertical axis ($Y$-axis). On her exam, she writes the following:

"To find the intersection with the $Y$-axis, we set $x = 0$:

$f(0) = \sqrt{0 - 4} + 3 = \sqrt{-4} + 3$.

Since the square root of a negative number does not exist in the real numbers, that part cancels out and is equal to $0$. Therefore, $f(0) = 0 + 3 = 3$. The graph intersects the $Y$-axis at the point $(0, 3)$".

1. a) Explain the conceptual error that Sofía makes when manipulating the square root of $-4$.
2. b) Find the domain of definition of $f(x)$ and use it to justify whether the graph intersects the $Y$-axis or not.

We want to determine the formula of a radical function of the form $f(x) = \sqrt{x - h} + v$ from a series of conditions on its graph:

1. Its domain of definition starts exactly at $x = 3$; that is, the function only exists for $x \geq 3$.

2. The curve intersects the horizontal axis ($X$-axis) at the point $(7, 0)$.

1. a) From the domain condition, deduce with reasoning the value of the parameter $h$.
2. b) Substitute the coordinates of the intersection point with the $X$-axis into the expression to calculate the exact value of $v$.
3. c) Write the complete analytical expression of the function $f(x)$ and describe what horizontal and vertical translations it has undergone with respect to the parent function $y = \sqrt{x}$.

Determine the domain of definition of the function $f(x) = \sqrt{-x^2 + 2x + 8}$.

The period $T$ (in seconds) of a simple pendulum is given by the function $T(l) = 2\pi\sqrt{\frac{l}{9,8}}$, where $l$ is the length of the pendulum in metres. Calculate the period of a pendulum of length $2,5\text{ m}$ and find out what length a pendulum must have so that its period is exactly $2\text{ s}$.

Look at the graph of the radical function $f(x)$. From it, determine its domain, its range, the axis intercepts, and its intervals of increase or decrease.

Find analytically the points of intersection between the radical function $f(x) = \sqrt{2x + 3}$ and the line $g(x) = x$.

The propagation speed $v$ (in m/s) of a tsunami in the ocean depends on the depth $d$ (in metres) according to the function $v(d) = \sqrt{9,8 \cdot d}$. If a tsunami travels at a speed of $720\text{ km/h}$, what is the depth of the ocean at that point? (Remember to make any necessary unit conversions).

Determine the inverse function of the right-hand branch of the parabola $f(x) = x^2 - 4$ (that is, for $x \ge 0$). What is the domain of the resulting inverse function?

Find the domain of definition of the function $g(x) = \frac{5}{\sqrt{x^2 - 16}}$.

A lighthouse is located on a straight coastline. A boat sails parallel to the coast at a distance of $3\text{ km}$ from it (passing right in front of the lighthouse at its closest point). If the boat moves away at a constant speed of $15\text{ km/h}$, write the function that determines the straight-line distance from the boat to the lighthouse as a function of time $t$ (in hours).

To find out whether the graph of a radical function intersects the horizontal axis ($X$-axis), Marcos and Elena propose two different methods:

• Marcos's method (algebraic): Set up the equation $f(x) = 0$, isolate the radical, square both sides, and check whether the obtained solution is valid.

• Elena's method (graphical and range): Analyze the vertex or starting point $(h, v)$ and observe the vertical shift $v$ and the growth of the curve to deduce directly whether it crosses the $X$-axis or not.

1. a) Apply Elena's method to justify, without solving any equation, why the function $g(x) = \sqrt{x - 1} + 4$ can never intersect the $X$-axis.
2. b) Apply Marcos's method to the function $f(x) = \sqrt{x + 5} - 3$ to find the exact coordinates of its intersection point with the $X$-axis.
3. c) Explain in what situation it is faster and more advantageous to use Elena's reasoning and when it is essential to resort to Marcos's algebraic method.

Accident investigation teams of the Guardia Civil de Tráfico in Spain use mathematical models to estimate the speed at which a car was travelling before performing an emergency stop on dry asphalt. Under standard conditions, the speed $v$ (in $\text{km/h}$) is related to the length of the skid mark $d$ (in metres) by the radical function:

1. a) On a section of a Spanish conventional road with a maximum speed limit of $90\text{ km/h}$, following an accident, a skid mark of $36\text{ metres}$ is measured. Calculate the estimated speed at which the car was travelling and determine whether it exceeded the legal speed limit.
2. b) Determine the mathematical domain of the function $v(d)$ and explain the physical meaning in this real context of not being able to input values of $d < 0$.
3. c) If a driver claims they were travelling at $70\text{ km/h}$, how many metres long should their skid mark have been? Solve the corresponding equation.

Starting from the basic radical function $f(x) = \sqrt{x}$, we apply the following geometric transformations in order: a shift of $3$ units to the right, a reflection across the x-axis, and finally, a shift of $4$ units up. Write the algebraic expression of the new function $g(x)$, calculate its domain and find its axis intercepts.

The following graph corresponds to a function of the type $f(x) = a\sqrt{x-b} + c$. Look at the key points marked on the curve and determine the exact value of the parameters $a$, $b$ and $c$.

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

A cable is to be stretched from the top of a tower of height $10\text{ m}$ to a point on the ground, and from there to a house located at $25\text{ m}$ from the base of the tower. If we call $x$ the distance from the base of the tower to the anchoring point on the ground, write the function $L(x)$ representing the total length of the cable.

Find the domain of definition of the function $f(x) = \sqrt{\frac{2x - 6}{x + 4}}$.

Solve analytically the system formed by the functions $y = \sqrt{x + 4}$ and $y = \sqrt{x - 1} + 1$ to find their point of intersection.

According to the Special Theory of Relativity, time dilates for an object moving at high speed. The Lorentz factor $\gamma$ is given by the function $\gamma(v) = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $v$ is the speed of the object and $c \approx 300\,000\text{ km/s}$ is the speed of light. Determine the mathematical domain of this function and briefly explain its physical meaning in terms of the maximum achievable speed.

Study and determine the domain of definition of the function $h(x) = \sqrt{|x - 3| - 2}$.

Express the area of an isosceles triangle inscribed in a circle of radius $R = 5\text{ cm}$ in terms of the height $h$ of the triangle. What is the mathematical domain of this function in its geometric context?

In maritime navigation off the Spanish coast, the maximum visual distance $D$ (in kilometres) at which the horizon line can be seen from an observation height $h$ (in metres above sea level) is modeled by the function:

An observer goes up to the viewpoint of the Chipiona Lighthouse (Cádiz), located $64\text{ metres}$ above sea level.

1. a) Estimate: Without using a calculator, mentally round the coefficient $3{,}57$ to $3{,}5$ and calculate an approximate estimate of the distance in kilometres to the visible horizon from that height. Explain your mental calculation in words.
2. b) Verify: Now perform the precise calculation with the exact value from the formula ($3{,}57$) and assess whether your previous estimate was reasonable and useful for maritime navigation.
3. c) Proportional reasoning: If the observer were to double their height by going up in a helicopter to $128\text{ metres}$ above sea level, will the distance at which they see the horizon also double? Justify your answer by analyzing the properties of the radical function without simply giving the numerical result.

The Confederación Hidrográfica del Ebro oversees discharge operations at Spanish dams. According to Torricelli's law, the water outflow velocity $v$ (in metres per second, $\text{m/s}$) through a deep gate depends on the water surface height $h$ (in metres) above the reservoir bed by means of the radical function:

where $5$ is subtracted because the first $5\text{ metres}$ of the base correspond to the safety zone for sludge settling, below which the gates remain closed.

1. a) Find the domain of definition of this function and interpret its practical meaning for the technical staff managing the dam.
2. b) During a drought period, the water level drops from $41\text{ metres}$ to $14\text{ metres}$. A technician claims that, since the height has dropped to almost a third, the velocity of the jet will also be reduced to one third. Check algebraically whether the technician's statement is correct or incorrect by calculating the velocities in both cases.
3. c) To ensure the optimal operation of the associated hydroelectric turbines, the water outflow velocity must be at least $14\text{ m/s}$. Set up and solve the corresponding equation to determine the minimum height $h$ at which the reservoir must be.