Functions with Roots
A hardware store in Santiago calculates the time it takes an object to fall from a bridge using a formula that contains a square root. If the value under the root comes out negative, the result does not exist as a real number: that situation makes no sense. How can you know, before calculating, for which values of $x$ the formula works? That is finding the domain, and here you will learn to do it exactly, even in complicated cases.
Domain with even index: solve the inequality of the radicand
If $g(x)$ is quadratic, the inequality does not always give a half-line.
Radicand with a parameter: when is a trinomial never negative?
Sometimes the radicand contains a letter (a parameter $k$) and you are asked for which values of $k$ the domain is all of $\mathbb{R}$. That means the trinomial $ax^2+bx+c$ must be $\geq 0$ for every $x$. The graph of the trinomial is a parabola, and the discriminant $\Delta=b^2-4ac$ tells whether it crosses the $x$-axis.
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:
$f(x) = \frac{1}{\sqrt[4]{x^2 - 5x + 6}}$
Determine the values of the parameter $k$ for which the domain of the function $f(x) = \sqrt{x^2 + kx + 9}$ is the entire set of real numbers $\mathbb{R}$.
Determine the values of the parameter $k$ for which the domain of the function $f(x) = \sqrt{x^2 + kx + 9}$ is the entire set of real numbers $\mathbb{R}$.
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
Inequalities with a root: $\sqrt{A(x)}\geq B(x)$ and $\sqrt{A(x)}\leq B(x)$
When the nested root has more than $x$ inside, as in $\sqrt{\sqrt{x+7}-x-1}$, you have to solve $\sqrt{x+7}\geq x+1$. Here you cannot square without thinking: it depends on the sign of the right-hand side $B(x)$.
Root multiplied by $x$: sign study in a profit problem
When the root is multiplied by a variable, you study the sign of each factor. Look at this case of a distributor: the revenue is $I(x)=x\sqrt{36-x}$ and the cost is $C(x)=3x$, in thousands of pesos, with $x$ tons sold. When is there a profit?
When the cost has a constant term (for example $C(x)=2x+5$), it doesn't factor so cleanly. Then the practical approach is to compute $B(x)$ at the value the problem asks for: if it comes out positive, there is a profit; if negative, a loss.
When the radicand is a fraction of polynomials: factor first
For the sign chart you need the roots of the numerator and of the denominator. If the polynomial is of degree 3 or higher, you have to factor it. Two tools:
-
Factoring by grouping: group terms in pairs and take out the common factor.
-
Synthetic division (Ruffini's rule): try divisors of the constant term; if $p(a)=0$, then $(x-a)$ is a factor.
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:
$f(x) = \sqrt{x + 4} + \sqrt{6 - 2x}$
Determine the domain of the irrational function whose radicand is an algebraic fraction with a quadratic numerator: $f(x) = \sqrt{\frac{x^2 - 9}{x - 1}}$
Determine the domain of the irrational function whose radicand is an algebraic fraction with a quadratic numerator:
$f(x) = \sqrt{\frac{x^2 - 9}{x - 1}}$
Find the domain of definition of the nested function: $f(x) = \sqrt{2 - \sqrt{x - 3}}$
Find the domain of definition of the nested function:
$f(x) = \sqrt{2 - \sqrt{x - 3}}$
Determine the domain of the irrational function containing a rational expression where the numerator and denominator share roots:…
Determine the domain of the irrational function containing a rational expression where the numerator and denominator share roots:
$f(x) = \sqrt{\frac{x^3 - 3x^2 - 4x + 12}{x^2 + x - 2}}$
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 you swap $x$ and $y$ and solve for the new $y$. Since a function with an even root is not always one-to-one on its whole natural domain, sometimes you have to restrict it first.
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 $f(x) = 3 + \sqrt{2x - 4}$, determine its inverse function $f^{-1}(x)$ and state the domain and range of this inverse function.
Given the function $f(x) = 3 + \sqrt{2x - 4}$, determine its inverse function $f^{-1}(x)$ and state the domain and range of this inverse function.
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 have to solve for the second, a root appears: oscillation periods, speeds, radii from areas. Here the domain must also make sense in context: a length or an area cannot be negative.
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 meters. 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.
The production cost $C(x)$, in pesos, of $x$ parts of an electronic component for satellites follows a model of diminishing returns given by…
The production cost $C(x)$, in pesos, of $x$ parts of an electronic component for satellites follows a model of diminishing returns given by $C(x) = 150\,000 + 7\,200\sqrt{x - 50}$. Determine the valid domain for the cost function and calculate how many parts were manufactured if the total cost amounted to $294\,000$ pesos.
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 $3 \times 10^8$ 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.
A rectangle is inscribed in the region bounded by the x-axis and the upper semicircle of radius $R = 5$, whose equation is $y = \sqrt{25 - x^2}$. Express the perimetre…
A rectangle is inscribed in the region bounded by the x-axis and the upper semicircle of radius $R = 5$, whose equation is $y = \sqrt{25 - x^2}$. Express the perimetre of the rectangle $P(x)$ in terms of the coordinate $x$ of the vertex located in the first quadrant. Determine the valid domain of $P(x)$ in this geometric context and calculate its value for $x = 3$.
An engineering company estimates that the daily revenue $I(x)$ and costs $C(x)$, in thousands of pesos, when manufacturing $x$ metric tons of a composite material are…
An engineering company estimates that the daily revenue $I(x)$ and costs $C(x)$, in thousands of pesos, when manufacturing $x$ metric tons of a composite material are given by $I(x) = x\sqrt{40 - x}$ and $C(x) = 2x + 5$. Determine the domain of the profit function $B(x) = I(x) - C(x)$ in the context of the problem and calculate whether the company makes a profit or a loss when manufacturing $15$ metric tons.
Go back to the beginning: the hardware store's formula only works for the values that leave the radicand greater than or equal to zero, and solving that inequality gives you the exact domain.