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 meters) can be approximated by the function $t(h) = \sqrt{\frac{h}{5}}$. How long will a ball dropped from a bridge $45$ meters high take to reach the ground?

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 meters. 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 meters) of the water, according to the function $v(d) = 11.2 \sqrt{d}$. If a tsunami has been detected traveling 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.

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 meters) above sea level can be approximated by the function $d(h) = 3.57 \sqrt{h}$. If you are at the top of a lighthouse $10$ meters high, how many additional meters 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 pesos) of an electronic components factory depends on the number of units produced $x$ according to the function $C(x) = 4\,000 + 200\sqrt{x - 100}$. Determine the domain of the function in this context. If the cost of a batch was $7\,000$ pesos, 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).