Functions with roots
An engineer calculates the natural vibration frequency of a beam using a formula that contains a square root, under which there is an expression involving the stiffness and the mass of the material. If that expression turns out negative for certain values, the formula makes no physical sense: that combination of material and geometry does not exist. In Bachillerato it is not enough to say "the radicand cannot be negative": you have to solve the corresponding inequality rigorously, however complicated it is, and give the exact domain as an interval.
Domain with even index: solving the inequality for the radicand
If $g(x)$ is quadratic, the inequality does not always give a half-line.
Determine the domain of definition of the following function with an even-index root: $f(x) = \sqrt{3x - 12}$
Determine the domain of definition of the following function with an even-index root:
$f(x) = \sqrt{3x - 12}$
Find the domain of the quadratic function under a square root: $h(x) = \sqrt{16 - x^2}$
Find the domain of the quadratic function under a square root:
$h(x) = \sqrt{16 - x^2}$
Calculate the domain of the function containing an even-index root in the denominator: $f(x) = \frac{1}{\sqrt[4]{x^2 - 5x + 6}}$
Calculate the domain of the function containing an even-index root in the denominator:
$f(x) = \frac{1}{\sqrt[4]{x^2 - 5x + 6}}$
Domain with odd index: no restriction
Calculate the domain of definition of the function with an odd-index root: $g(x) = \sqrt[3]{2x^2 - 8x + 1}$
Calculate the domain of definition of the function with an odd-index root:
$g(x) = \sqrt[3]{2x^2 - 8x + 1}$
Domain in compound expressions: fractions, sums and nested roots
Find the domain of the irrational function containing a rational expression: $f(x) = \sqrt{\frac{x - 2}{x + 5}}$
Find the domain of the irrational function containing a rational expression:
$f(x) = \sqrt{\frac{x - 2}{x + 5}}$
Find the domain of definition of the function formed by the sum of two roots: $f(x) = \sqrt{x + 4} + \sqrt{6 - 2x}$
Find the domain of definition of the function formed by the sum of two roots:
$f(x) = \sqrt{x + 4} + \sqrt{6 - 2x}$
Solve the inequality associated with the domain of the function $f(x) = \sqrt{\sqrt{x + 7} - x - 1}$ and express it in interval notation.
Solve the inequality associated with the domain of the function $f(x) = \sqrt{\sqrt{x + 7} - x - 1}$ and express it in interval notation.
Shape of the graph, range and translations
From the function $f(x) = \sqrt{x}$, write the algebraic expression for the function $g(x)$ whose graph is obtained by translating $f(x)$ three units to the left and two…
From the function $f(x) = \sqrt{x}$, write the algebraic expression for the function $g(x)$ whose graph is obtained by translating $f(x)$ three units to the left and two units upwards. What is the domain of $g(x)$?
Determine the domain and range of the function with a negative leading coefficient: $f(x) = -\sqrt{x - 1}$
Determine the domain and range of the function with a negative leading coefficient:
$f(x) = -\sqrt{x - 1}$
Observe the attached graph representing a function of the form $f(x) = a\sqrt{x - b} + c$. Determine the values of the parameters $a$, $b$ and $c$, knowing that the…
Observe the attached graph representing a function of the form $f(x) = a\sqrt{x - b} + c$. Determine the values of the parameters $a$, $b$ and $c$, knowing that the vertex of the semi-parabola is located at the point $(2, 1)$ and it passes through the point $(6, -3)$.
Intercepts, intersections and the inverse function
To find the inverse function, swap $x$ and $y$ and solve for the new $y$. Since a function with an even-index root is not always one-to-one over all of $\mathbb{R}$, sometimes you first have to restrict its domain.
Find the axis intercepts of the function: $f(x) = \sqrt{x + 9} - 3$
Find the axis intercepts of the function:
$f(x) = \sqrt{x + 9} - 3$
Calculate the points of intersection between the irrational function $f(x) = \sqrt{2x + 5}$ and the linear function $g(x) = x + 1$.
Calculate the points of intersection between the irrational function $f(x) = \sqrt{2x + 5}$ and the linear function $g(x) = x + 1$.
Given the function $f(x) = \sqrt{x^2 - 4x}$, calculate its inverse function $f^{-1}(x)$ by restricting the original domain to the interval $(-\infty, 0]$. Clearly state…
Given the function $f(x) = \sqrt{x^2 - 4x}$, calculate its inverse function $f^{-1}(x)$ by restricting the original domain to the interval $(-\infty, 0]$. Clearly state the domain and range of the resulting inverse function.
Functions with roots in science and technology
Whenever one quantity depends on the square of another and you need to solve for the second, a root appears: oscillation periods, terminal velocities, radii from areas or volumes.
The relationship between the area $A$ of a circle and its radius $r$ is given by the formula $A = \pi r^2$. Express the radius in terms of the area, $r(A)$, and state…
The relationship between the area $A$ of a circle and its radius $r$ is given by the formula $A = \pi r^2$. Express the radius in terms of the area, $r(A)$, and state the valid domain of definition in this geometric context.
The period $T$ (in seconds) of a simple pendulum can be approximated by the function $T(L) = 2\pi\sqrt{\frac{L}{9,8}}$, where $L$ is the length of the pendulum in…
The period $T$ (in seconds) of a simple pendulum can be approximated by the function $T(L) = 2\pi\sqrt{\frac{L}{9,8}}$, where $L$ is the length of the pendulum in metres. Calculate the length required for the period to be exactly $2$ seconds and state the theoretical domain of the function $T(L)$ in this physical context.
In the theory of special relativity, the Lorentz factor $\gamma$ depends on the velocity $v$ of an object according to the function…
In the theory of special relativity, the Lorentz factor $\gamma$ depends on the velocity $v$ of an object according to the function $\gamma(v) = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $c$ is the speed of light in a vacuum (approximately $3 \cdot 10^8$ m/s). Determine the domain of the function $\gamma(v)$ and interpret physically what happens to the function as $v$ approaches the boundaries of this domain.