Radical Functions
You type $\sqrt{5-8}$ into your graphing calculator and it answers ERR: NONREAL ANS. The calculator is telling you that $\sqrt{-3}$ is not a real number. Engineers, physicists, and economists run into the same wall: a formula with a root only makes sense for some inputs. In this section you will find those inputs exactly, by solving an inequality, and you will learn how to graph, transform, and invert radical functions. At the end we come back to the calculator error.
Even Index: Solve the Radicand Inequality
Why? An even power of a real number is never negative, so no real number squared (or raised to the 4th power) gives a negative result. A negative radicand has no real root.
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:
$f(x) = \sqrt{3x - 12}$
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:
$f(x) = \frac{1}{\sqrt[4]{x^2 - 5x + 6}}$
Quadratic Radicands and a Parameter
When the radicand is quadratic, the answer is not always a half-line. It depends on whether the parabola $y=g(x)$ opens up or down and where it crosses the x-axis.
When is the domain all of $\mathbb{R}$?
Sometimes a parameter is hiding in the radicand, as in $\sqrt{x^2+kx+9}$. The domain is all real numbers exactly when $x^2+kx+9\ge 0$ for every $x$. An upward parabola never dips below the x-axis when it has no x-intercepts or just touches it. That means the discriminant is not positive: $$b^2-4ac\le 0$$
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:
$h(x) = \sqrt{16 - x^2}$
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 $\mathbb{R}$.
Odd Index: No Restriction
An odd power can be negative (just cube $-2$), so every real number has a real odd root.
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:
$g(x) = \sqrt[3]{2x^2 - 8x + 1}$
Rational Radicands: Sign Charts
Why a sign chart? You cannot just multiply both sides by $q(x)$, because you do not know whether $q(x)$ is positive or negative. A sign chart avoids the problem.
The method:
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Find the zeros of the numerator and of the denominator. These are the critical points.
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Place them on a number line and test each interval.
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Include a zero of the numerator (the quotient is $0$). Never include a zero of the denominator.
When the numerator and denominator share factors
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:
$f(x) = \sqrt{\frac{x - 2}{x + 5}}$
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:
$f(x) = \sqrt{\frac{x^2 - 9}{x - 1}}$
Determine the domain of the irrational function containing a rational expression where the numerator and the denominator share roots:…
Determine the domain of the irrational function containing a rational expression where the numerator and the denominator share roots:
$f(x) = \sqrt{\frac{x^3 - 3x^2 - 4x + 12}{x^2 + x - 2}}$
Sums of Roots and Nested Roots
Why intersection? To add two numbers, both must exist at the same time. If one root is not real at some $x$, the sum is not a real number there.
A root inside an inequality: $\sqrt{A}\ge B$
For a function like $\sqrt{\sqrt{x+2}-x}$ you must solve $\sqrt{x+2}\ge x$. Split into two cases based on the sign of the side without the root:
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If $B<0$: the inequality $\sqrt{A}\ge B$ is automatically true wherever $\sqrt{A}$ exists, because a root is never negative.
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If $B\ge 0$: both sides are nonnegative, so you may square: $A\ge B^2$.
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:
$f(x) = \sqrt{x + 4} + \sqrt{6 - 2x}$
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:
$f(x) = \sqrt{2 - \sqrt{x - 3}}$
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.
Graphs, Range, and Transformations
Some problems call these irrational functions and the graph a semi-parabola: it is half of a parabola lying on its side. We just call them radical functions.
Why does it flatten? To raise $y$ by one unit at a time, $x$ must jump $0,1,4,9,16,\dots$ Each step needs a bigger jump in $x$, so the curve bends toward the right.
| $x$ | 0 | 1 | 4 | 9 | 16 |
|---|---|---|---|---|---|
| $\sqrt{x}$ | 0 | 1 | 2 | 3 | 4 |
Use these friendly points to graph transformations. For $f(x)=a\sqrt{x-b}+c$:
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$b$ shifts the graph horizontally: right if $b>0$, left if $b<0$. The starting point moves to $x=b$.
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$c$ shifts it vertically.
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$a$ stretches or compresses it. If $a<0$ it also reflects the graph over the horizontal line through the starting point.
Why? The root is always $\ge 0$. Multiplying by $a<0$ makes it $\le 0$, so after adding $c$ the outputs are at most $c$, and the vertex is the highest point.
From the function $f(x) = \sqrt{x}$, write the analytic expression of 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 analytic expression of 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 the range of the function with a negative leading coefficient: $f(x) = -\sqrt{x - 1}$
Determine the domain and the 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 passes through the point $(6, -3)$.
Intercepts and Intersections
To find where a radical graph meets another graph, set the two expressions equal. Squaring is the key step, and it can create extraneous solutions.
Find the points of intersection with the coordinate axes of the function: $f(x) = \sqrt{x + 9} - 3$
Find the points of intersection with the coordinate axes 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 $g(x) = x + 1$.
Inverse of a Radical Function
To find $f^{-1}$: write $y=f(x)$, swap $x$ and $y$, and solve for the new $y$. Remember that the domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.
When the radicand is quadratic
A function like $\sqrt{x^2-2x}$ is not one-to-one on its whole domain (the graph is symmetric about $x=1$), so you must first restrict it to one side.
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 inverse function obtained.
Radicals as Models
Whenever one quantity depends on the square of another and you solve for the second, a root appears: the radius from an area, the time of a fall, the swing time of a pendulum.
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 feet.…
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 feet. Calculate the length needed 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 dollars, of $x$ pieces of an electronic component for satellites follows a model of diminishing returns given by…
The production cost $C(x)$, in dollars, of $x$ pieces of an electronic component for satellites follows a model of diminishing returns given by $C(x) = 2500 + 120\sqrt{x - 50}$. Determine the valid domain for the cost function and calculate how many pieces have been manufactured if the total cost has amounted to $\$4900$.
Domain in Context and Behavior at the Endpoints
In a modeling problem the domain comes from two sources: the algebra (where the formula exists) and the situation (what values make sense). Take the intersection. Then look at what the function does near each endpoint.
For a profit function $B(x)=I(x)-C(x)$, the domain is the intersection of the domains of $I$ and $C$, intersected with whatever the context requires (for example $x\ge 0$). To decide profit or loss at a given $x$, compute $B(x)$: positive means profit, negative means loss.
What happens near an endpoint?
In the theory of special relativity, the Lorentz factor $\gamma$ depends on the speed $v$ of an object according to the function…
In the theory of special relativity, the Lorentz factor $\gamma$ depends on the speed $v$ 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 $3 \cdot 10^8$ ft/s). Determine the domain of the function $\gamma(v)$ and interpret physically what happens to the function as $v$ approaches the endpoints 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 perimeter…
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 perimeter 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 dollars, when manufacturing $x$ tons of a composite material are given…
An engineering company estimates that the daily revenue $I(x)$ and costs $C(x)$, in thousands of dollars, when manufacturing $x$ 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 $15$ tons.
Back to the start: your calculator said ERR: NONREAL ANS for $\sqrt{5-8}$ because the radicand $-3$ is negative, so $5-8$ falls outside the domain of an even-index root. Now you can find exactly which inputs the calculator would accept.