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{{ printedBook.courseTrack.name }} {{ printedBook.name }} A radical function is a function that contains a radical expression, where the independent variable is under the radical sign. One example is

$f(x) = 2\sqrt{x + 1}.$When the radical of a radical function is a square root, it is called a square root function. The parent function of the square root function family is $f(x) = \sqrt{x}.$

Note that the square root of a negative number is imaginary. Thus, when not allowing for non-real numbers, the domain of the parent function is all real numbers greater than or equal to $0.$ The square root of a positive number is also always positive, which leads to the range being all real numbers greater than or equal to $0.$When the radical is a cube root instead, it is called a cube root function. The parent function of the family of cube root functions is $f(x) = \sqrt[3]{x}.$

Note that the cube root of a negative number is defined. Thus, both its domain and range are all real numbers.With the help of tables, graph the given functions. $f(x) = \sqrt{x + 1} \qquad g(x) = \sqrt[3]{2x}$ Then, for both functions, give their following features:

- end behavior,
- symmetry, if any,
- increasing and decreasing interval(s).

Show Solution

Let's start by graphing $f.$ This is a square root function, so it is not defined when the expression inside the radical is negative. Thus, it is best to start our table with $x = \text{-} 1,$ which leads to $0$ under the radical.

$x$ | $\sqrt{x + 1}$ | $f(x)$ |
---|---|---|

${\color{#0000FF}{\text{-} 1}}$ | $\sqrt{{\color{#0000FF}{\text{-} 1}} + 1}$ | $0$ |

${\color{#0000FF}{0}}$ | $\sqrt{{\color{#0000FF}{0}} + 1}$ | $1$ |

${\color{#0000FF}{1}}$ | $\sqrt{{\color{#0000FF}{1}} + 1}$ | $\sim 1.41$ |

${\color{#0000FF}{2}}$ | $\sqrt{{\color{#0000FF}{2}} + 1}$ | $\sim 1.73$ |

${\color{#0000FF}{3}}$ | $\sqrt{{\color{#0000FF}{3}} + 1}$ | $2$ |

We can now plot the corresponding points $(x, f(x))$ in a coordinate plane. Connecting the points with a smooth curve gives us the desired graph. Note that the function is not defined for $x$ less than $\text{-} 1.$

Notice that $g$ is a cube root function — it is defined for all values of $x.$ Thus, we'll use both positive and negative values for the table.

$x$ | $\sqrt[3]{2x}$ | $g(x)$ |
---|---|---|

${\color{#0000FF}{\text{-} 2}}$ | $\sqrt[3]{2({\color{#0000FF}{\text{-} 2}})}$ | $\sim \text{-} 1.59$ |

${\color{#0000FF}{\text{-} 1}}$ | $\sqrt[3]{2({\color{#0000FF}{\text{-} 1}})}$ | $\sim \text{-} 1.26$ |

${\color{#0000FF}{0}}$ | $\sqrt[3]{2({\color{#0000FF}{0}})}$ | $0$ |

${\color{#0000FF}{1}}$ | $\sqrt[3]{2({\color{#0000FF}{1}})}$ | $\sim 1.26$ |

${\color{#0000FF}{2}}$ | $\sqrt[3]{2({\color{#0000FF}{2}})}$ | $\sim 1.59$ |

Let's plot the points and connect them with a smooth curve. Since the domain of this function is all real numbers, the graph should continue infinitely in both directions.

To determine the stated key features, we'll analyze each graph. Both these functions extend upward as $x$ extends to positive infinity. However, their left-ends differ, as $f$ is not defined for $x$ less than $\text{-} 1,$ while $g$ extends downward. $\begin{aligned} \mathbf{f(x)} \quad \text{left-end} &: \ f(x) \rightarrow \text{-} 1 \quad && \text{right-end} : \ f(x) \rightarrow + \infty \\ \mathbf{g(x)} \quad \text{left-end} &: \ g(x) \rightarrow \text{-} \infty \quad && \text{right-end} : \ g(x) \rightarrow + \infty \\ \end{aligned}$ For a function to have even or odd symmetry, it has to fulfill $f(\text{-} x) = f(x) \quad \text{or} \quad f(\text{-} x) = \text{-} f(x),$ respectively. Graphically, this is a symmetry in the $y$-axis or about the origin, respectively. $f$ has neither of these, while $g$ has odd symmetry — opposite sign of the input yields opposite sign of the output. Assuming no other behavior of the functions than that seen in the graphs, both functions are increasing for their entire domain.

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