Exercises: Radical functions

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

Given the function $f(x) = \sqrt{2x - 1}$, calculate its values for $x = 5$ and $x = 13$.

Find the domain of the function $g(x) = \sqrt{-3x + 12}$.

Calculate the axes intercepts of the function $h(x) = \sqrt{x + 4} - 2$.

Look at the three functions represented in the following graph. Identify which of them corresponds to the expression $f(x) = \sqrt{x - 2}$ and justify your answer by stating its domain.

Calculate the value of $x$ for which the function $f(x) = \sqrt{5x + 1}$ takes the value $4$.

Determine the domain of the function $f(x) = 3 - \sqrt{x + 5}$.

The time $t$ (in seconds) that an object takes to fall from a height $h$ (in metres) can be approximated by the function $t(h) = \sqrt{\frac{h}{5}}$. How long will a ball dropped from a bridge $45$ metres high take to reach the ground?

Lucía tries to calculate the $y$-intercept of the function $f(x) = \sqrt{x - 4} + 1$. To do so, she writes in her notebook: “Since the $y$-intercept occurs when $x = 0$, I calculate $f(0) = \sqrt{0 - 4} + 1 = \sqrt{-4} + 1 = -2 + 1 = -1$, so the curve intersects at the point $(0, -1)$”. Identify the two conceptual errors made by Lucía, explain why her procedure contradicts the definition of the domain of a radical function in real numbers, and determine, with justification, whether the graph of $f$ intersects the $y$-axis or not.

An elementary radical function has the expression $f(x) = \sqrt{x - h} + v$. It is known that its graphical representation starts exactly at the point $(-3, 2)$ in the Cartesian plane and that it also passes through the point $(1, 4)$. Determine in a reasoned manner the numerical values of the parameters $h$ and $v$. From them, deduce the domain and range of the function relying on the meaning of horizontal and vertical translations with respect to the basic function $y = \sqrt{x}$.

The period $T$ (in seconds) of a simple pendulum is given by the function $T(L) = 2\pi \sqrt{\frac{L}{10}}$, where $L$ is the length of the string in metres. If we want to construct a pendulum that has a period of exactly $2$ seconds, what length should the string have? (Use $\pi \approx 3,14$).

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

Given the function $f(x) = \sqrt{x}$, write the analytical expression of the function $g(x)$ resulting from translating the graph of $f(x)$ three units to the right and two units upwards.

The speed $v$ (in km/h) at which a tsunami propagates in the ocean depends on the depth $d$ (in metres) of the water, according to the function $v(d) = 11,2 \sqrt{d}$. If a tsunami has been detected travelling at a speed of $336$ km/h, what is the depth of the ocean at that point?

Calculate the axes intercepts of the function $f(x) = \sqrt{2x + 3} - 3$.

The radius $r$ of a circle in terms of its area $A$ is given by $r(A) = \sqrt{\frac{A}{\pi}}$. Calculate the average rate of change of the radius as the area increases from $100\pi$ to $400\pi$ square units.

Determine the values of the parameters $a$ and $b$ so that the function $f(x) = \sqrt{ax + b}$ has the interval $[-2, +\infty)$ as its domain and passes through the point $(2, 4)$.

Look at the graphs of the functions $f(x) = \sqrt{x + 1}$ and $g(x) = \sqrt{-x + 1}$. Determine the domain of each of them and calculate the exact point where they intersect.

From the Cabo Vilán lighthouse (Galicia), the maximum theoretical distance $D$ (in kilometres) at which the horizon line over the sea can be seen depends on the height $h$ (in metres) of the observer's eyes, according to the model $D(h) = 3,57 \cdot \sqrt{h}$.

1. a) Before calculating the exact value, estimate in a reasoned manner between which two consecutive integer numbers of kilometres the viewing distance will lie for a visitor located at a height of $h = 40\text{ m}$ above sea level, justifying your approximation based on the closest perfect squares to $40$.
2. b) Now calculate the numerical value of $D(40)$ rounding to two decimal places and check whether your initial estimate was correct.
3. c) If a sailor claims that to see twice as far ($2 \cdot D$) it is enough to stand at twice the height ($2 \cdot h$), justify mathematically whether his claim is true or false by analysing how the square root behaves when the variable is doubled.

The State Meteorological Agency (AEMET) records wind speed $v$ (in $\text{m/s}$) in the vicinity of a wind farm in Aragón as a function of the height $z$ (in metres) above the ground using the expression $v(z) = 3 + 2\sqrt{z - 4}$, applicable starting from the minimum height of the anemometers.

1. a) Interpret the mathematical and physical meaning of the radicand and deduce the domain of definition of the function in this real-world context. Why does it make no sense to evaluate $v(2)$?
2. b) Set up the corresponding equation to find at what height the wind speed reaches $11\text{ m/s}$, solve it step by step, and explicitly check whether the solution obtained belongs to the domain of the function.
3. c) Without solving any long algebraic equation, deduce in a reasoned manner from the range (image) of the function whether it is possible that at some point above the minimum height the wind is completely calm ($v = 0\text{ m/s}$).

Determine the domain of definition of the function $f(x) = \sqrt{-x^2 + 6x - 8}$.

Find the points of intersection between the line $y = x - 2$ and the radical function $y = \sqrt{x + 4}$. Check that the solution obtained belongs to the domain of both functions and satisfies the original equation.

The distance $d$ (in km) to the horizon for an observer situated at a height $h$ (in metres) above sea level can be approximated by the function $d(h) = 3,57 \sqrt{h}$. If you are at the top of a lighthouse $10$ metres high, how many additional metres would you have to climb so that the distance to the visible horizon doubles?

Determine the domain and the range of the function $f(x) = -2 + \sqrt{3x - 6}$. Justify your answer by analysing the characteristics of the radical function.

The production cost $C$ (in euros) of an electronic components factory depends on the number of units produced $x$ according to the function $C(x) = 200 + 10\sqrt{x - 100}$. Determine the domain of the function in this context. If the cost of a batch was $350$ euros, how many units were produced?

The relationship between the volume $V$ of a cube and its edge $a$ is $V = a^3$. However, if we consider the area of one of its faces $A = a^2$, express the edge $a$ in terms of the area $A$. Then, express the volume $V$ in terms of the area $A$ using a radical function and calculate the volume if the area of one face is $25 \text{ cm}^2$.

Given the function $f(x) = \sqrt{x^2 - 4x}$, determine the values of $x$ for which the function is defined. Then, calculate the exact points where the function takes the value $\sqrt{5}$.

Look at the following piecewise function and its graphical representation:

$f(x) = \begin{cases} \sqrt{-x} & \text{if } x \le 0 \\ \sqrt{x} & \text{if } x > 0 \end{cases}$

Find its domain, reason whether it is a continuous function at $x=0$ and prove algebraically whether it possesses any type of symmetry (even or odd).

To find the intersection points between the radical function $f(x) = \sqrt{2x + 10}$ and the line $g(x) = x - 1$, one must solve the equation $\sqrt{2x + 10} = x - 1$. Two students propose different strategies before operating:

• Marcos's method: Isolate the square root, square both sides, solve the resulting quadratic equation, and, at the end, substitute each solution into the starting equality to discard possible extraneous solutions.

• Sara's method: Before squaring, analyse the domain of the radicand ($2x + 10 \ge 0$) and the sign condition of the second member ($x - 1 \ge 0$), deducing that any valid solution must necessarily belong to the interval $[1, +\infty)$.

1. a) Apply Marcos's method by solving the equation step by step. Show which extraneous solution arises and explain algebraically why it appears when squaring.
2. b) Explain why Sara's method allows discarding the extraneous solution immediately without the need to substitute it into the original equation.
3. c) Justify what danger there is in squaring directly if the initial equation were presented non-isolated as $\sqrt{2x + 10} - x = -1$ and the square were applied to each term separately.

Design in a reasoned manner the analytical expression of a radical function of the form $f(x) = a\sqrt{x - h} + v$ that simultaneously satisfies these four conditions:

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

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

3. The function is strictly decreasing throughout its domain.

4. The maximum value that the function reaches is $6$ (that is, its range is $(-\infty, 6]$).

Answer the following questions with justification:

1. a) Deduce step by step the value of each of the parameters $h$, $v$, and $a$, explaining which condition allows you to determine each one.
2. b) Determine analytically whether the graph of your function intersects the $y$-axis or not.
3. c) If we change only the sign of the coefficient $a$ obtained (leaving $h$ and $v$ untouched), what would the new range of the function be and what would happen to the $x$-intercept? Justify your answer without merely stating the result.