Radical functions
An object dropped from a height $h$ takes $t(h) = \sqrt{h/5}$ seconds to reach the ground. This function has a peculiarity that straight lines and parabolas did not have: it is not defined for every $h$. There are no negative heights, and there is no square root of a negative number either. Before using a radical function you must always find out where it lives.
The domain: the radicand cannot be negative
Determine the domain of definition of the function $f(x) = \sqrt{x - 8}$.
Determine the domain of definition of the function $f(x) = \sqrt{x - 8}$.
Find the domain of the function $g(x) = \sqrt{-3x + 12}$.
Find the domain of the function $g(x) = \sqrt{-3x + 12}$.
Determine the domain of the function $f(x) = 3 - \sqrt{x + 5}$.
Determine the domain of the function $f(x) = 3 - \sqrt{x + 5}$.
Calculating images
To find $f(a)$ you substitute $x=a$ into the radicand and work out the root. If the radicand turns out negative, $a$ does not belong to the domain: the image does not exist.
Given the function $f(x) = \sqrt{2x - 1}$, calculate its values for $x = 5$ and $x = 13$.
Given the function $f(x) = \sqrt{2x - 1}$, calculate its values for $x = 5$ and $x = 13$.
Calculate the value of $x$ for which the function $f(x) = \sqrt{5x + 1}$ takes the value $4$.
Calculate the value of $x$ for which the function $f(x) = \sqrt{5x + 1}$ takes the value $4$.
Translations of $y=\sqrt{x}$
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…
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.
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…
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 $(2, 4)$.
Intercepts with the axes
With the $Y$-axis: calculate $f(0)$, if $0$ is in the domain. With the $X$-axis: solve $f(x)=0$, squaring both sides to eliminate the root.
Calculate the axes intercepts of the function $h(x) = \sqrt{x + 4} - 2$.
Calculate the axes intercepts of the function $h(x) = \sqrt{x + 4} - 2$.
Calculate the axes intercepts of the function $f(x) = \sqrt{2x + 3} - 3$.
Calculate the axes intercepts of the function $f(x) = \sqrt{2x + 3} - 3$.
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…
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.
Radical functions in real-life contexts
The square root appears whenever one quantity depends on the square of another and we want to solve for the second: periods of pendulums, speeds of propagation, distances to the horizon.
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…
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?
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…
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 $\pi \approx 3,14$).
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 $v(d) = 11,2 \sqrt{d}$.…
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?
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 $d(h) = 3,57 \sqrt{h}$.…
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?
The production cost $C$ (in euros) of an electronic components factory depends on the number of units produced $x$ according to the 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?
Back to the beginning: In the falling-object function $t(h) = \sqrt{h/5}$, the condition $h/5 \geq 0$ requires $h \geq 0$: the mathematical and physical domain is $[0, +\infty)$, which confirms that it makes no sense to calculate times for negative heights.