Radical Functions
If you drop a ball from a bridge of height $h$ (in meters), it takes $t(h)=\sqrt{h/5}$ seconds to reach the ground. This function has something that lines and parabolas did not: it is not defined for every $h$. There are no negative heights, and the square root of a negative number does not exist either. Before using a radical function you must always find out where it exists.
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 $f(x) = 3 - \sqrt{x + 5}$.
Domain with a quadratic radicand
When the radicand is second-degree you have to solve a quadratic inequality. The method: factor, find the roots, and check the sign in each region.
If the parabola of the radicand opens downward, the domain is the interval between the roots. For example, for $\sqrt{9-x^2}$ we need $9-x^2\geq 0$, which gives $[-3,3]$.
Find the domain of definition of the function $f(x) = \sqrt{x^2 - 16}$.
Find the domain of definition of the function $f(x) = \sqrt{x^2 - 16}$.
Determine the domain of definition of the function $f(x) = \sqrt{-x^2 + 6x - 8}$.
Determine the domain of definition of the function $f(x) = \sqrt{-x^2 + 6x - 8}$.
Calculating images and given values
To find $f(a)$, substitute $x=a$ into the radicand and compute the root. If the radicand turns out negative, $a$ is not in the domain and the image does not exist.
For the reverse direction (which $x$ gives a certain output?), set up an equation and square both sides. For example, if $\sqrt{2x-1}=7$, then $2x-1=49$ and $x=25$.
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 $y=\sqrt{x}$
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…
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.
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 $(2, 4)$.
The range
The range of a function is the set of all values that $y$ can take. A principal root is never negative, so $\sqrt{x-h}$ takes values from $0$ upward. Adding $v$ outside shifts everything $v$ units.
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.
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.
Average rate of change
The average rate of change of $f$ between $x=a$ and $x=b$ measures how much $f$ changes per unit of $x$: $$\frac{f(b)-f(a)}{b-a}$$ It is the slope of the line joining the points $(a, f(a))$ and $(b, f(b))$.
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…
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.
Intercepts with the axes
With the $Y$-axis: compute $f(0)$, if $0$ is in the domain. With the $X$-axis: solve $f(x)=0$, isolating the root and squaring.
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 equations and intersections
To find where two graphs cross, set their expressions equal. If one is radical: isolate the root, square, solve, and check each solution.
When two radical functions are required to take the same value, you can also set the insides equal, as long as both radicands are in the domain. And if you are asked for the value $\sqrt{5}$, the radicand must equal $5$ (and, along the way, must be $\geq 0$).
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…
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.
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…
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 $\sqrt{5}$.
Radical functions in real-world contexts
The square root appears when one quantity depends on the square of another and we want to solve for the latter: periods of pendulums, wave speeds, distances to the horizon.
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…
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…
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 $\pi \approx 3.14$).
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 $v(d) = 11.2 \sqrt{d}$.…
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?
Models with restrictions
In a model, the mathematical domain comes with one from the context: there are no negative heights and you don't produce $-3$ units. To solve for the variable you have to "undo" the root with a square, and to go from one variable to another you compose: if $a=\sqrt{A}$, then anything that depends on $a$ also depends on $A$.
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 $d(h) = 3.57 \sqrt{h}$.…
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?
The production cost $C$ (in pesos) of an electronic components factory depends on the number of units produced $x$ according to the 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…
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 $25 \text{ cm}^2$.
Continuity and symmetry in piecewise functions
A function is continuous at a point if its graph can be drawn there without lifting your pencil: the two pieces that meet give the same value.
A function is even if $f(-x)=f(x)$ for every $x$ in the domain (its graph is symmetric about the $Y$-axis), and odd if $f(-x)=-f(x)$ (symmetric about the origin). For either to hold, the domain must be symmetric about $0$.
Look at the following piecewise function and its graphical representation:…
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).
Back to the beginning: in the falling function $t(h)=\sqrt{h/5}$ the radicand requires $h/5\geq 0$, that is, $h\geq 0$: the domain is $[0,+\infty)$, exactly what the physical meaning requires. For $h=45$ m, $t=\sqrt{9}=3$ s.