Algebra 2 · F.IF.C.7B

Cube root functions and roots as powers

A square root turns away every negative input. What does a cube root do with them, and what could a power of one third mean?

Where you are 3 in Radical functions

Functions13

Radical functions3

Graphing square root functions Radical functions in context

Why this exists

A cube of side s holds a volume of s^3. Often the question runs the other way: a box must hold 3375 cubic centimeters, so how long is its edge? Undoing a cube is taking a cube root, and turning that undoing into a rule that accepts an input and returns an output gives the cube root function.

It sits in the family of radical functions beside the square root, but it behaves differently. No real number squares to a negative, so \displaystyle \sqrt{x} refuses every negative input and its graph is a half curve that starts at the origin. A negative number does have a cube root, since (-2)^3 = -8. So the cube root accepts every real input, and its graph runs across the whole page, through the origin, in both directions.

There is a second way to write the same function. Exponents already have rules, such as (a^m)^n = a^{mn}. If a power of one third is to obey that rule, then \left(x^{1/3}\right)^3 = x^1 = x, which says x^{1/3} must be a number whose cube is x. So a power of one third is a cube root, and a root function is a power function in disguise.

Definition

The cube root function is

f(x) = \sqrt[3]{x}

and its output is the one real number whose cube is the input:

y = \sqrt[3]{x} \quad\text{means}\quad y^3 = x

So \displaystyle \sqrt[3]{8} = 2 because 2^3 = 8, and \displaystyle \sqrt[3]{-8} = -2 because (-2)^3 = -8. The small 3 is the index of the root and the expression under the sign is the radicand. The square root \displaystyle \sqrt{x} has index 2, which is left unwritten; in general \displaystyle \sqrt[n]{x}, the nth root of x, is a number whose nth power is x.

The key points of the cube root graph are at inputs that are perfect cubes:

x	-8	-1	0	1	8
\displaystyle \sqrt[3]{x}	-2	-1	0	1	2

Figure 1The cube root takes every input, so its graph runs across the whole page. The square root (dashed) takes no negative input, so its graph is only a half.

Roots as powers. For a whole number n \ge 2,

x^{1/n} = \sqrt[n]{x}

for every real x when n is odd, and for x \ge 0 when n is even. So the cube root function is also the power function y = x^{1/3}, and the square root function is y = x^{1/2}. The denominator of the exponent is the index of the root. Brackets matter: (5x)^{1/3} is \displaystyle \sqrt[3]{5x}, while 5x^{1/3} is \displaystyle 5\sqrt[3]{x}, because a power applies only to what it is written on.

A transformed cube root function has the form

y = a\sqrt[3]{x - h} + k, \qquad a \ne 0

where, as in graphing by transforming a parent graph, h moves the parent graph sideways, k moves it up or down, and a stretches it vertically (and flips it when a is negative). The point (h, k), where the origin of the parent lands, is the center point.

Intuition

The cube root undoes cubing, so its graph is the cubing graph turned on its side. The point (2, 8) is on y = x^3 because 2^3 = 8; the same fact puts (8, 2) on \displaystyle y = \sqrt[3]{x}. Every point (a, b) of one graph becomes the point (b, a) of the other. Swapping the two coordinates reflects a point in the line y = x, so the two graphs are mirror images in that line.

Figure 2Swapping the two coordinates of every point of y = x^3 gives \displaystyle y = \sqrt[3]{x}. The swap is a reflection in the line y = x.

Steep near 0, slow far away. The cubing graph of the squaring and cubing functions lies flat at the origin and climbs faster and faster. Turned on its side, the cube root graph stands upright at the origin and then climbs more and more slowly. Between 0 and 1 a cube root is bigger than its input, \displaystyle \sqrt[3]{0.001} = 0.1; past 1 it is much smaller, \displaystyle \sqrt[3]{1000} = 10. To double the output, the input must grow eightfold.

The sign comes through unchanged. Positive inputs have positive cube roots and negative inputs negative ones, with \displaystyle \sqrt[3]{-x} = -\sqrt[3]{x}. A half turn about the origin carries the graph onto itself, the symmetry about the origin described in symmetry about the axis and the origin.

Results

The cube root function \displaystyle y = \sqrt[3]{x}:

has domain all real numbers and range all real numbers, written (-\infty, \infty) in Interval and set-builder notation;
passes through (-8, -2), (-1, -1), (0, 0), (1, 1) and (8, 2);
rises everywhere: if a < b then \displaystyle \sqrt[3]{a} < \sqrt[3]{b}, so each output comes from exactly one input;
undoes cubing and is undone by it: \displaystyle \sqrt[3]{x^3} = x and \displaystyle \left(\sqrt[3]{x}\right)^3 = x for every real x;
is the mirror image of y = x^3 in the line y = x;
satisfies \displaystyle \sqrt[3]{-x} = -\sqrt[3]{x}, so its graph is symmetric about the origin;
stands vertical for an instant at the origin: it has no slope there;
satisfies \displaystyle \sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b} for all real a and b.

Roots as powers. \displaystyle x^{1/n} = \sqrt[n]{x}, for all real x when n is odd and for x \ge 0 when n is even. Every odd root \displaystyle y = \sqrt[n]{x} has the cube root’s shape across the whole page through (-1, -1), (0, 0) and (1, 1); every even root has the square root’s half shape.

Transformed graphs. For a \ne 0, \displaystyle y = a\sqrt[3]{x - h} + k has domain and range all real numbers, center point (h, k), and key points (h - 8, k - 2a), (h - 1, k - a), (h, k), (h + 1, k + a) and (h + 8, k + 2a). It rises everywhere when a > 0 and falls everywhere when a < 0, and it meets every horizontal line exactly once.

Why it is true

Every real number has exactly one real cube root. The cubing function rises everywhere and reaches every height, as shown in the squaring and cubing functions. So each height c is reached at exactly one input, and that input is \displaystyle \sqrt[3]{c}. This is why no input is refused: the domain of the cube root is the range of cubing, all real numbers. And since every real x is the cube of some number (its own cube root), every real number is an output, so the range is all real numbers too.

It rises. Suppose a < b but \displaystyle \sqrt[3]{a} \ge \sqrt[3]{b}. Cubing rises, so cubing both sides keeps the order: a \ge b, a contradiction. So \displaystyle \sqrt[3]{a} < \sqrt[3]{b}.

The mirror. The point (a, b) lies on y = x^3 exactly when b = a^3, which is exactly when \displaystyle a = \sqrt[3]{b}, which is exactly when (b, a) lies on \displaystyle y = \sqrt[3]{x}.

The sign. If r^3 = x then (-r)^3 = -r^3 = -x, since three negative factors give a negative product. So -r is the cube root of -x: \displaystyle \sqrt[3]{-x} = -\sqrt[3]{x}.

The product rule. By the power of a product,

\begin{aligned} \left(\sqrt[3]{a}\,\sqrt[3]{b}\right)^3 &= \left(\sqrt[3]{a}\right)^3 \left(\sqrt[3]{b}\right)^3 \\ &= ab \end{aligned}

So \displaystyle \sqrt[3]{a}\,\sqrt[3]{b} is a real number whose cube is ab, and there is only one such number, \displaystyle \sqrt[3]{ab}.

Upright at the origin. Take the point (t^3, t) on the graph, with t \ne 0. The segment from the origin to it has slope \displaystyle \frac{t}{t^3} = \frac{1}{t^2}, which grows without limit as t shrinks toward 0. The segments stand up more and more steeply, so the curve is vertical at the origin, and a vertical line has no slope.

Powers. The rule (a^m)^n = a^{mn} of powers of powers holds for whole number exponents. If it is to hold for \displaystyle m = \frac{1}{n} as well, then \left(x^{1/n}\right)^n = x, so x^{1/n} has to be a number whose nth power is x. When n is odd there is exactly one real such number for every x, for the same reason as for cubes. When n is even, an nth power is never negative, so only x \ge 0 can work, and the non-negative choice is the one named \displaystyle \sqrt[n]{x}.

Transformed graphs. Replacing x by x - h moves a graph h to the right, multiplying the output by a stretches it vertically by a, and adding k moves it up by k (shifting a graph, stretches, reflections). Each of these sends every horizontal line’s single meeting point to a single meeting point, so the result still meets every horizontal line once. Putting x - h = -8, -1, 0, 1, 8 gives \displaystyle \sqrt[3]{x - h} = -2, -1, 0, 1, 2, which are the key points listed.

Worked examples

Example 1

Domain and values

Let \displaystyle q(x) = \sqrt[3]{5x - 2}. Find its domain, then q(2), q(-5) and q(0.4). Compare the domain with that of \displaystyle \sqrt{5x - 2}.

Solution

The index is 3, which is odd, so the radicand 5x - 2 may be any real number, negative included. Every x gives a real radicand, so the domain of q is all real numbers.

Substitute and look for a perfect cube under the sign.

\displaystyle q(2) = \sqrt[3]{10 - 2} = \sqrt[3]{8} = 2, since 2^3 = 8.

\displaystyle q(-5) = \sqrt[3]{-25 - 2} = \sqrt[3]{-27} = -3, since (-3)^3 = -27.

\displaystyle q(0.4) = \sqrt[3]{2 - 2} = \sqrt[3]{0} = 0.

For \displaystyle \sqrt{5x - 2} the index is even, so the radicand must not be negative: 5x - 2 \ge 0 gives x \ge 0.4. That function has no value at x = -5 at all; the cube root has the value -3 there.

Example 2

Where the brackets matter

Evaluate each, or say why it is not a real number: (-343)^{1/3}, -343^{1/3}, 0.008^{1/3}, (-81)^{1/4} and -81^{1/4}.

Solution

Each power of \displaystyle \frac{1}{n} is an nth root. A power applies only to what it is written on, so in -343^{1/3} the exponent belongs to 343 and the minus sign is applied afterwards.

\displaystyle (-343)^{1/3} = \sqrt[3]{-343} = -7, since (-7)^3 = -343.

\displaystyle -343^{1/3} = -\sqrt[3]{343} = -7. The two readings agree because an odd root lets the sign pass straight through: \displaystyle \sqrt[3]{-x} = -\sqrt[3]{x}.

\displaystyle 0.008^{1/3} = \sqrt[3]{0.008} = 0.2, since 0.2^3 = 0.008.

\displaystyle (-81)^{1/4} = \sqrt[4]{-81} is not a real number: a fourth power is a square of a square, never negative, so no real number has fourth power -81.

\displaystyle -81^{1/4} = -\sqrt[4]{81} = -3, since 3^4 = 81. For an even root the two readings no longer agree, so the brackets decide the answer.

Example 3

Graphing the lead function

Graph \displaystyle f(x) = \sqrt[3]{x + 2} - 3. Give its center point, five key points, its domain and range, and both intercepts.

Solution

Write the rule in the form \displaystyle a\sqrt[3]{x - h} + k: here x + 2 = x - (-2), so h = -2, k = -3 and a = 1. The parent graph moves 2 left and 3 down, and the center point is (-2, -3).

Key points come from making the radicand a perfect cube. Setting x + 2 = -8, -1, 0, 1, 8 gives x = -10, -3, -2, -1, 6, and the outputs are -2 - 3, -1 - 3, 0 - 3, 1 - 3 and 2 - 3:

x	-10	-3	-2	-1	6
f(x)	-5	-4	-3	-2	-1

Figure 3The graph of \displaystyle f(x) = \sqrt[3]{x + 2} - 3 (solid) is the parent (dashed) moved 2 left and 3 down, so its center point is (-2, -3).

Shifting does not change the fact that every input is allowed and every height is reached, so the domain and the range are both all real numbers.

The x-intercept is where f(x) = 0: \displaystyle \sqrt[3]{x + 2} = 3. Cubing both sides gives x + 2 = 27, so x = 25 and the intercept is (25, 0). Check: \displaystyle \sqrt[3]{27} - 3 = 0.

The y-intercept is \displaystyle f(0) = \sqrt[3]{2} - 3 \approx -1.74, the point \displaystyle (0, \sqrt[3]{2} - 3). The graph rises slowly, which is why it takes until x = 25 to climb from -3 to 0.

Example 4

A stretch that is also a squeeze

Graph \displaystyle g(x) = 2\sqrt[3]{x - 1} from its key points. Then show that \displaystyle \sqrt[3]{8(x - 1)} is the same function.

Solution

Here a = 2, h = 1 and k = 0, so the center point is (1, 0) and every output of the parent is doubled. Setting x - 1 = -8, -1, 0, 1, 8 gives x = -7, 0, 1, 2, 9 and outputs twice -2, -1, 0, 1, 2:

x	-7	0	1	2	9
g(x)	-4	-2	0	2	4

Figure 4The graph of \displaystyle g(x) = 2\sqrt[3]{x - 1} (solid) is the parent (dashed) moved 1 right and stretched upwards by a factor of 2.

The x-intercept is the center point (1, 0), and the y-intercept is \displaystyle g(0) = 2\sqrt[3]{-1} = -2.

By the product rule for cube roots, which holds for every real radicand,

\sqrt[3]{8(x - 1)} = \sqrt[3]{8}\,\sqrt[3]{x - 1} = 2\sqrt[3]{x - 1}

So squeezing the parent sideways by a factor of 8 gives the same graph as stretching it upwards by a factor of 2. For a cube root graph, a horizontal squeeze and a vertical stretch are two descriptions of one move.

Example 5

Finding the rule from two points

A cube root graph of the form \displaystyle y = a\sqrt[3]{x - h} + k has center point (2, 1) and passes through (10, -3). Find its rule and check it against the point (1, 3).

Solution

The center point gives h = 2 and k = 1, so \displaystyle y = a\sqrt[3]{x - 2} + 1. Only a is unknown. Substituting the point (10, -3):

-3 = a\sqrt[3]{10 - 2} + 1 = a\sqrt[3]{8} + 1 = 2a + 1

so 2a = -4 and a = -2. The rule is \displaystyle y = -2\sqrt[3]{x - 2} + 1. The negative a means the graph falls from left to right.

Check at x = 1: \displaystyle -2\sqrt[3]{-1} + 1 = -2(-1) + 1 = 3, so (1, 3) is on the graph.

Example 6

Solving and undoing

For the lead function \displaystyle f(x) = \sqrt[3]{x + 2} - 3, find the input with f(x) = -7. Then find the inverse function f^{-1}.

Solution

Set \displaystyle \sqrt[3]{x + 2} - 3 = -7 and add 3 to both sides: \displaystyle \sqrt[3]{x + 2} = -4. Cubing both sides gives x + 2 = -64, so x = -66. Check: \displaystyle \sqrt[3]{-64} - 3 = -4 - 3 = -7.

Cubing both sides is safe here in a way that squaring is not. Cubing rises everywhere, so two different numbers never have the same cube, and the cubed equation has exactly the solutions of the original. No answer has to be thrown out.

For the inverse, write \displaystyle y = \sqrt[3]{x + 2} - 3 and solve for x: \displaystyle y + 3 = \sqrt[3]{x + 2}, then (y + 3)^3 = x + 2, so x = (y + 3)^3 - 2. Renaming the input,

f^{-1}(x) = (x + 3)^3 - 2

Check: f^{-1}(-7) = (-4)^3 - 2 = -66, matching the first part. Because f meets every horizontal line exactly once, it has an inverse on its whole domain, with no need to cut the domain down first as the squaring function needs.

Example 7

Boxes that hold twice as much

Ileana makes cube-shaped storage boxes. A box that holds V cubic centimeters has edge \displaystyle s(V) = \sqrt[3]{V} centimeters. Find the edge of a box holding 3375 cubic centimeters, of one holding twice that, and of one holding eight times that.

Solution

\displaystyle s(3375) = \sqrt[3]{3375} = 15 centimeters, since 15^3 = 3375.

Twice the volume is 6750. By the product rule, \displaystyle s(6750) = \sqrt[3]{2}\,\sqrt[3]{3375}, which is \displaystyle 15\sqrt[3]{2} \approx 18.9 centimeters. Doubling the volume makes the edge only about 26\% longer, because \displaystyle \sqrt[3]{2} \approx 1.26.

Eight times the volume is 27000, and \displaystyle s(27000) = \sqrt[3]{8}\,\sqrt[3]{3375}, which is 2 \cdot 15 = 30 centimeters. In general s(8V) = 2s(V): the edge doubles only when the volume grows eightfold, which is the slow climb of the cube root graph seen in a box.

Boundaries

Only odd roots accept every input. \displaystyle \sqrt[3]{x}, \displaystyle \sqrt[5]{x} and every odd root take all real numbers. An even root such as \displaystyle \sqrt{x} or \displaystyle \sqrt[4]{x} takes only x \ge 0; that half graph is the subject of graphing square root functions.

One real cube root, not three. Among complex numbers, -8 has three cube roots, and only one of them, -2, is real. The sign \displaystyle \sqrt[3]{\ } in this lesson always means the real one.

Fractional powers of negative numbers. Writing (-8)^{1/3} = -2 follows the definition above, but the usual exponent rules can break once the base is negative. The fraction \displaystyle \frac{1}{3} equals \displaystyle \frac{2}{6}, yet \left((-8)^2\right)^{1/6} = 64^{1/6} = 2, not -2. Rules such as (a^m)^n = a^{mn} with fractional exponents are safe for a \ge 0 only. For the same reason some calculators and programming languages refuse a power such as (-8)^{1/3} or return a complex number; writing \displaystyle -\sqrt[3]{8} avoids the question.

No slope at the center. The graph is vertical for an instant at its center point, so it has no slope there, though it has one everywhere else.

The stretch must not be zero. With a = 0, \displaystyle y = a\sqrt[3]{x - h} + k is the horizontal line y = k, which is not a cube root graph and does not reach every height.

No edge and no ceiling. The graph climbs ever more slowly, but it never levels off: for any height c, the input c^3 reaches it. It has no endpoint, unlike the square root, and no horizontal asymptote.

Where this goes next

Powers of one third and one half are the start of rational exponents, where x^{2/3} means \displaystyle \left(\sqrt[3]{x}\right)^2 and all the exponent rules carry over to fractions. The cube root’s place as the reverse of cubing is one case of inverse functions, and its square root sibling needs a domain cut in half before it can play the same role. The moves used on the graph here are gathered in combining transformations, and they work the same way on square root graphs. Quantities that grow like a root, such as the side of a cube or the radius of a ball of given volume, appear in radical functions in context.

Exercises

Values and domains

Let \displaystyle r(x) = \sqrt[3]{x}. Find r(-216), r(0.064), \displaystyle r\left(-\frac{125}{8}\right) and r(1000).
Let \displaystyle k(x) = \sqrt[3]{2x + 7}. Find k(10), k(-7.5) and k(-3.5).
Give the domain of each function: (a) \displaystyle \sqrt[3]{6 - 4x}; (b) \displaystyle \sqrt{6 - 4x}; (c) \displaystyle \sqrt[5]{x^2 - 9}; (d) \displaystyle \frac{1}{\sqrt[3]{x - 5}}.
Find the range of \displaystyle y = \sqrt[3]{x} when the inputs are restricted to -27 \le x \le 64.

Roots as powers

Evaluate 729^{1/3}, (-0.027)^{1/3} and 625^{1/4}.
Evaluate each, or say why it is not a real number: (-1000)^{1/3}, (-25)^{1/2}, -25^{1/2} and (-1)^{1/7}.
Write with a fractional exponent: \displaystyle \sqrt[3]{9w}, \displaystyle 4\sqrt[3]{w} and \displaystyle \sqrt[7]{w}.
Find 125^{1/3} \cdot 125^{1/3}.

Graphs and transformations

For \displaystyle y = \sqrt[3]{x - 4} + 2, give the center point, five key points and both intercepts.
For \displaystyle y = -\sqrt[3]{x} + 1, give five key points, say whether the graph rises or falls, and give both intercepts.
For \displaystyle y = 3\sqrt[3]{x + 5}, give the center point, five key points and the y-intercept, to two decimal places.
Find both intercepts of \displaystyle y = \sqrt[3]{x - 7} - 1, the y-intercept to three decimal places.
Describe how the graph of \displaystyle y = \frac{1}{2}\sqrt[3]{x + 6} - 4 is made from \displaystyle y = \sqrt[3]{x}, and give its center point and five key points.
A graph of the form \displaystyle y = a\sqrt[3]{x - h} + k has center point (-1, 4) and passes through (0, 1). Find its rule, and its value at x = 7.
Write \displaystyle y = \sqrt[3]{27x} as a vertical stretch of \displaystyle y = \sqrt[3]{x}.
Write \displaystyle y = \sqrt[3]{-x} without a minus sign under the root, and say what this means for the two reflections of the graph.

Equations, inverses and uses

Solve \displaystyle \sqrt[3]{3x - 1} = -2.
Solve \displaystyle 2\sqrt[3]{x - 1} = 5.
Find the inverse of h(x) = x^3 + 9.
Brannoc’s spherical water tank has volume \displaystyle V = \frac{4}{3}\pi r^3, so its radius is \displaystyle r = \sqrt[3]{\frac{3V}{4\pi}}. (a) Find the radius when V = 288\pi cubic meters. (b) By what factor must the volume grow for the radius to double?

Reasoning

Explain why \displaystyle \sqrt[3]{x^3} = x for every real x, while \displaystyle \sqrt{x^2} = x fails at x = -5.
Prove that \displaystyle \sqrt[3]{x} > x for 0 < x < 1 and \displaystyle \sqrt[3]{x} < x for x > 1, and say what this means for the two graphs.
An argument runs: (-27)^{1/3} = (-27)^{2/6}, which is \left((-27)^2\right)^{1/6}, which is 729^{1/6} = 3. But (-27)^{1/3} = -3. Which step fails, and why?
Explain why, when a \ne 0, every horizontal line meets the graph of \displaystyle y = a\sqrt[3]{x - h} + k exactly once, and why no such statement holds for \displaystyle y = a\sqrt{x - h} + k.

Answers

r(-216) = -6, r(0.064) = 0.4, \displaystyle r\left(-\frac{125}{8}\right) = -\frac{5}{2}, r(1000) = 10. Check by cubing: (-6)^3 = -216, 0.4^3 = 0.064, \displaystyle \left(-\frac{5}{2}\right)^3 = -\frac{125}{8}, 10^3 = 1000.
\displaystyle k(10) = \sqrt[3]{27} = 3, \displaystyle k(-7.5) = \sqrt[3]{-8} = -2, \displaystyle k(-3.5) = \sqrt[3]{0} = 0.
(a) All real numbers: the index is odd, so a negative radicand is fine. (b) 6 - 4x \ge 0, so x \le 1.5. (c) All real numbers: the index 5 is odd. (d) Every real number except x = 5: the cube root accepts every input, but at x = 5 it is 0 and the fraction would divide by zero.
The function rises, so the smallest output is at the smallest input and the largest at the largest: \displaystyle \sqrt[3]{-27} = -3 and \displaystyle \sqrt[3]{64} = 4. The range is -3 \le y \le 4.
729^{1/3} = 9 since 9^3 = 729; (-0.027)^{1/3} = -0.3 since (-0.3)^3 = -0.027; 625^{1/4} = 5 since 5^4 = 625.
(-1000)^{1/3} = -10. (-25)^{1/2} is not a real number, since no real square is negative. \displaystyle -25^{1/2} = -\sqrt{25} = -5. (-1)^{1/7} = -1, since (-1)^7 = -1.
\displaystyle \sqrt[3]{9w} = (9w)^{1/3}, with brackets because the whole radicand is under the root; \displaystyle 4\sqrt[3]{w} = 4w^{1/3}, since only w is under the root; \displaystyle \sqrt[7]{w} = w^{1/7}.
125^{1/3} \cdot 125^{1/3} = 5 \cdot 5 = 25.
Center point (4, 2). Key points (-4, 0), (3, 1), (4, 2), (5, 3), (12, 4), from x - 4 = -8, -1, 0, 1, 8. The x-intercept is (-4, 0), since \displaystyle \sqrt[3]{-8} + 2 = 0. The y-intercept is \displaystyle \sqrt[3]{-4} + 2 \approx 0.413.
Key points (-8, 3), (-1, 2), (0, 1), (1, 0), (8, -1). Here a = -1, so the graph falls everywhere. The x-intercept is (1, 0) and the y-intercept is (0, 1), the center point.
Center point (-5, 0). Key points (-13, -6), (-6, -3), (-5, 0), (-4, 3), (3, 6). The y-intercept is \displaystyle 3\sqrt[3]{5} \approx 5.13.
x-intercept: \displaystyle \sqrt[3]{x - 7} = 1, so x - 7 = 1 and x = 8. y-intercept: \displaystyle \sqrt[3]{-7} - 1 = -\sqrt[3]{7} - 1 \approx -2.913.
Move 6 left, squash vertically by a factor of \displaystyle \frac{1}{2}, move 4 down. Center point (-6, -4). Key points (-14, -5), (-7, -4.5), (-6, -4), (-5, -3.5), (2, -3).
\displaystyle y = a\sqrt[3]{x + 1} + 4, and (0, 1) gives \displaystyle 1 = a\sqrt[3]{1} + 4, so a = -3: \displaystyle y = -3\sqrt[3]{x + 1} + 4. At x = 7: \displaystyle -3\sqrt[3]{8} + 4 = -2.
\displaystyle \sqrt[3]{27x} = \sqrt[3]{27}\,\sqrt[3]{x} = 3\sqrt[3]{x}, a vertical stretch by a factor of 3.
\displaystyle \sqrt[3]{-x} = -\sqrt[3]{x}. Reflecting the graph in the y-axis gives the same graph as reflecting it in the x-axis.
Cubing both sides, 3x - 1 = -8, so \displaystyle x = -\frac{7}{3}. Check: \displaystyle \sqrt[3]{-7 - 1} = \sqrt[3]{-8} = -2.
\displaystyle \sqrt[3]{x - 1} = 2.5, so x - 1 = 2.5^3 = 15.625 and x = 16.625.
Write y = x^3 + 9, so x^3 = y - 9 and \displaystyle x = \sqrt[3]{y - 9}: \displaystyle h^{-1}(x) = \sqrt[3]{x - 9}. Check: \displaystyle \sqrt[3]{(x^3 + 9) - 9} = \sqrt[3]{x^3} = x.
(a) \displaystyle \frac{3V}{4\pi} = \frac{3 \cdot 288\pi}{4\pi} = 216, so \displaystyle r = \sqrt[3]{216} = 6 meters. (b) The volume must grow by a factor of 8: doubling r multiplies r^3, and so V, by 2^3 = 8.
\displaystyle \sqrt[3]{x^3} asks for the one real number whose cube is x^3, and x is such a number, whatever its sign. A square root, though, is defined to be the non-negative number whose square is the radicand. At x = -5, \displaystyle \sqrt{(-5)^2} = \sqrt{25} = 5, not -5; in general \displaystyle \sqrt{x^2} = |x|.
Let \displaystyle r = \sqrt[3]{x}, so x = r^3, and r > 0 when x > 0. If 0 < x < 1 then 0 < r < 1 (the cube root rises, and \displaystyle \sqrt[3]{1} = 1), and multiplying r < 1 by the positive numbers r and then r again gives r^3 < r^2 < r, so \displaystyle x < \sqrt[3]{x}. If x > 1 then r > 1 and the same steps give r^3 > r, so \displaystyle x > \sqrt[3]{x}. So the cube root graph lies above the line y = x between 0 and 1 and below it past 1, crossing it at (1, 1).
The step (-27)^{2/6} = \left((-27)^2\right)^{1/6} fails. The rule a^{mn} = (a^m)^n with a fractional exponent holds for a \ge 0. With a negative base, squaring first throws the sign away, and the even root that follows cannot bring it back.
The parent \displaystyle y = \sqrt[3]{x} meets every horizontal line once, because every height c is reached at the single input c^3. Shifting moves each meeting point without making or losing one, and multiplying by a \ne 0 only renames the heights, so the transformed graph also meets each line once. The square root graph has range y \ge 0 before any change, so after the moves it reaches only the heights on one side of k: a line on the other side meets it nowhere.