Graph of the Cubic Root Function

Oct 13, 2025 by pynkocean.com 5 views

Introduction

In mathematics, a cubic root function is a type of function that involves the cube root of a variable. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In this article, we will explore the graph of the cubic root function $f(x)=\sqrt[3]{x}$ and determine which points lie on the graph.

The Cubic Root Function

The cubic root function is defined as $f(x)=\sqrt[3]{x}$ . This function takes a value $x$ and returns its cube root. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because $3 \times 3 \times 3 = 27$ .

Graph of the Cubic Root Function

The graph of the cubic root function is a curve that passes through the point $(0,0)$ and has a positive slope. The graph is symmetric about the y-axis, meaning that if the point $(x,y)$ lies on the graph, then the point $(-x,y)$ also lies on the graph.

Evaluating the Points

Now, let's evaluate each of the given points to determine which ones lie on the graph of the cubic root function.

(-27,-3)

To determine if the point $(-27,-3)$ lies on the graph, we need to check if $f(-27)=-3$ . We can calculate the value of $f(-27)$ by substituting $-27$ into the function:

f(-27)=\sqrt[3]{-27}=-3

Since $f(-27)=-3$ , the point $(-27,-3)$ lies on the graph.

(-1,1)

To determine if the point $(-1,1)$ lies on the graph, we need to check if $f(-1)=1$ . We can calculate the value of $f(-1)$ by substituting $-1$ into the function:

f(-1)=\sqrt[3]{-1}=-1

Since $f(-1)=-1$ , not 1, the point $(-1,1)$ does not lie on the graph.

(0,0)

To determine if the point $(0,0)$ lies on the graph, we need to check if $f(0)=0$ . We can calculate the value of $f(0)$ by substituting $0$ into the function:

f(0)=\sqrt[3]{0}=0

Since $f(0)=0$ , the point $(0,0)$ lies on the graph.

(1,1)

To determine if the point $(1,1)$ lies on the graph, we need to check if $f(1)=1$ . We can calculate the value of $f(1)$ by substituting $1$ into the function:

f(1)=\sqrt[3]{1}=1

Since $f(1)=1$ , the point $(1,1)$ lies on the graph.

(3,27)

To determine if the point $(3,27)$ lies on the graph, we need to check if $f(3)=27$ . We can calculate the value of $f(3)$ by substituting $3$ into the function:

f(3)=\sqrt[3]{3} \neq 27

Since $f(3) \neq 27$ , the point $(3,27)$ does not lie on the graph.

Conclusion

In conclusion, the points that lie on the graph of the cubic root function $f(x)=\sqrt[3]{x}$ are:

$(-27,-3)$
$(0,0)$
$(1,1)$

These points satisfy the equation $f(x)=\sqrt[3]{x}$ , and therefore lie on the graph of the function. The other points do not satisfy the equation and therefore do not lie on the graph.

Final Answer

The final answer is:

$(-27,-3)$
$(0,0)$
$(1,1)$