Radical Functions
Jayden drops a tennis ball from the top of a 100-foot cliff. The fall time in seconds is $t(h) = \dfrac{\sqrt{h}}{4}$, where $h$ is the height in feet. Plugging in $h = 100$ gives $\dfrac{10}{4} = 2.5$ seconds. But when Jayden types $h = -100$ into his calculator, it shows an error. Why? In this section you will learn to find exactly which inputs a radical function accepts, how its graph behaves, and how to solve equations with radicals.
Even-Index Radicals: Solve the Radicand Inequality
Why? Square roots (and 4th roots, 6th roots…) of negative numbers are not real numbers, because no real number squared is negative.
When the radicand is quadratic, the inequality does not always give a half-line.
When the parabola opens upward, the answer flips to the outside. Use a sign chart: factor, find the zeros, and test one point in each interval.
Determine the domain of the following function with an even-index radical: $f(x) = \sqrt{3x - 12}$
Determine the domain of the following function with an even-index radical:
$f(x) = \sqrt{3x - 12}$
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 radical in the denominator: $f(x) = \frac{1}{\sqrt[4]{x^2 - 5x + 6}}$
Calculate the domain of the function containing an even-index radical in the denominator:
$f(x) = \frac{1}{\sqrt[4]{x^2 - 5x + 6}}$
Odd-Index Radicals: No Restriction
An odd root can still have other restrictions. A fraction needs a nonzero denominator, and an odd root in a denominator needs a nonzero radicand.
Calculate the domain of the function with an odd-index root: $g(x) = \sqrt[3]{2x^2 - 8x + 1}$
Calculate the domain of the function with an odd-index root:
$g(x) = \sqrt[3]{2x^2 - 8x + 1}$
Compound Expressions: Fractions, Sums, and Nested Radicals
Find the domain of the radical function containing a rational expression: $f(x) = \sqrt{\frac{x - 2}{x + 5}}$
Find the domain of the radical function containing a rational expression:
$f(x) = \sqrt{\frac{x - 2}{x + 5}}$
Determine the domain of the radical 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 radical function whose radicand is an algebraic fraction with a quadratic numerator:
$f(x) = \sqrt{\frac{x^2 - 9}{x - 1}}$
Find the domain of the nested function: $f(x) = \sqrt{2 - \sqrt{x - 3}}$
Find the domain of the nested function:
$f(x) = \sqrt{2 - \sqrt{x - 3}}$
Harder Domain Problems
Three techniques show up in the toughest problems: a parameter in the radicand, a radical inside an inequality, and a fraction with several critical points.
A parameter in the radicand
The domain of $f(x) = \sqrt{x^2 + kx + 16}$ is all of $\mathbb{R}$ exactly when $x^2 + kx + 16 \geq 0$ for every $x$. The parabola opens upward, so this happens when it never dips below the x-axis. In other words, it has at most one x-intercept, so the discriminant $b^2 - 4ac \leq 0$.
A radical inside the inequality
To solve $\sqrt{A} \geq B$, split into two cases. If $B < 0$, it is true whenever $A \geq 0$ (a root is never negative). If $B \geq 0$, both sides are nonnegative, so you may square both sides.
Several critical points, and shared factors
Factor the numerator and denominator completely, list all the zeros of both, and test every interval.
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 $\mathbb{R}$.
Determine the domain of the radical function containing a rational expression where the numerator and the denominator share roots:…
Determine the domain of the radical function containing a rational expression where the numerator and the denominator share roots:
$f(x) = \sqrt{\frac{x^3 - 3x^2 - 4x + 12}{x^2 + x - 2}}$
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.
Graph Shape, Range, and Translations
To move the graph of $y = \sqrt{x}$, replace $x$ by $x - b$ to shift right $b$ units (so a shift left 3 is $x + 3$), and add $c$ to shift up $c$ units.
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 up. 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
Not every radical function has an x-intercept. For $f(x) = \sqrt{x+1} + 2$, the equation $\sqrt{x+1} = -2$ has no solution, because a square root is never negative.
Find the x- and y-intercepts of the function: $f(x) = \sqrt{x + 9} - 3$
Find the x- and y-intercepts of the function:
$f(x) = \sqrt{x + 9} - 3$
Calculate the points of intersection between the radical function $f(x) = \sqrt{2x + 5}$ and the linear function $g(x) = x + 1$.
Calculate the points of intersection between the radical function $f(x) = \sqrt{2x + 5}$ and the linear function $g(x) = x + 1$.
Inverse of a Radical Function
To find an inverse function $f^{-1}$, swap $x$ and $y$ and solve for the new $y$. The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.
The curves $y = \sqrt{x-1}$ and $y = x^2 + 1$ are mirror images over the line $y = x$, but only the part of the parabola with $x \geq 0$ is the inverse. That restriction is the key idea.
Some functions are not one-to-one on their whole domain. Then you restrict the domain to one branch before inverting.
Now suppose you restricted $g$ to the left branch $(-\infty, 0]$ instead. The outputs of $g$ are still $\geq 0$ (a square root is never negative), so the domain of the inverse is again $[0, \infty)$. What changes is the inverse's output, which is the original input: now $y \leq 0$, so $y - 3 \leq -3 < 0$ and you take the negative root: $g^{-1}(x) = 3 - \sqrt{x^2 + 9}$, with range $(-\infty, 0]$. Check: $g(-2) = \sqrt{4 + 12} = 4$ and $g^{-1}(4) = 3 - \sqrt{25} = -2$ ✓.
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.
Radical Functions in Science and Geometry
Whenever one quantity depends on the square of another and you need to solve for the second, a root appears: falling times, pendulum periods, radii from areas.
When the radical has a coefficient such as $2\pi$ or a fraction under the root, isolate the radical first, then square. The same plan works for cost models like $C(x) = a + b\sqrt{x - c}$, where the domain also requires $x - c \geq 0$.
Back to the start: the ball dropped from 100 ft takes $\frac{\sqrt{100}}{4} = 2.5$ seconds. The calculator errors on $h = -100$ because $\sqrt{-100}$ is not a real number, and a negative height makes no sense for a fall anyway: the domain of $t(h)$ is $h \geq 0$ (or $h > 0$ in context).
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 in this geometric 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 $\$4900$.
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 $x = 3$.