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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

$f(x)=2x+1 .$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 $f(x)=x .$

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 $f(x)=3x .$

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)=x+1 g(x)=32x $ Then, for both functions, give their following features:

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

Show Solution

$x$ | $x+1 $ | $f(x)$ |
---|---|---|

$-1$ | $-1+1 $ | $0$ |

$0$ | $0+1 $ | $1$ |

$1$ | $1+1 $ | $∼1.41$ |

$2$ | $2+1 $ | $∼1.73$ |

$3$ | $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 $-1.$

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.

$x$ | $32x $ | $g(x)$ |
---|---|---|

$-2$ | $32(-2) $ | $∼-1.59$ |

$-1$ | $32(-1) $ | $∼-1.26$ |

$0$ | $32(0) $ | $0$ |

$1$ | $32(1) $ | $∼1.26$ |

$2$ | $32(2) $ | $∼1.59$ |

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.

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