The **index** of a radical function can be any real number, but the most common are square roots with an index of 2, like y=\sqrt{x}, which are called a **square root function**, and cube roots with an index of 3, like y=\sqrt[3]{x}, called a **cube root function**.

Consider the functions:

- f\left(x\right)=\sqrt{x}
- g\left(x\right)=\sqrt{x+4}
- h\left(x\right)=\sqrt{x}-2
- k\left(x\right)=\sqrt[3]{x}

- Which functions have a real output for an input of x=-3? x=0? x=5?
- Which functions have a domain of all real values of x?
- Which functions go through the point \left(0,0\right)?
- Which functions can have negative outputs?
- Match each function to one of these graphs:

Graph 1

Graph 2

Graph 3

Graph 4

For cube root functions, the function increases (or decreases) at a fast rate, then the rate of change slows around a point called an **inflection point**. In other words, the function continues increasing (or decreasing), but the rate is slower around the point of inflection.

Radical functions can be transformed similarly to any transformation of the parent function, y= af \left[b\left(x-h \right)\right] +k.

Square root | Cube root | |
---|---|---|

\text{Parent function:} | y= \sqrt{x} | y= \sqrt[3]{x} |

\text{Reflection across the }x\text{-axis:} | y=-\sqrt{x} | y=-\sqrt[3]{x} |

\text{Reflection across the }y\text{-axis:} | y=\sqrt{-x} | y=\sqrt[3]{-x} |

\text{Vertical stretch when } \left|a\right|>1 \\ \text{Vertical compression when } 0<\left|a\right|<1 \text{:} | y=a\sqrt{x} | y=a\sqrt[3]{x} |

\text{Horizontal compression when } \left|b\right|>1 \\ \text{Horizontal stretch when } 0<\left|b\right|<1 \text{:} | y=\sqrt{bx} | y=\sqrt[3]{bx} |

\text{Horizontal translation by } h \\ \text{Vertical translation by } k \text{:} | y=\sqrt{x-h} + k | y=\sqrt[3]{x-h} + k |

(h, k) \text{:} | \text{Endpoint} | \text{Point of inflection} |

The domain and range of the square root function will change with a reflection, or as h or k changes, while the domain and range of the cube root function will continue to be all real numbers.

Similarly, the absolute extremum of the square root function will change location when translated. If there are no reflections, the endpoint of the domain is an absolute minimum. If a vertical reflection occurs, it becomes an absolute maximum.

Square root functions do not have a relative extremum, and cube root functions have neither absolute nor relative extrema.

For f\left(x\right) = -\sqrt{x + 2},

a

Describe the transformation that occurred to y=\sqrt{x} to give f\left(x\right).

Worked Solution

b

Draw a graph of the function.

Worked Solution

c

Write the domain and range of f\left(x\right).

Worked Solution

d

Determine the intervals where f\left(x\right) is increasing, decreasing, or constant.

Worked Solution

Consider the graph of f \left( x \right).

a

Write the equation that represents f\left(x\right).

Worked Solution

b

Write the domain and range of f\left(x\right).

Worked Solution

c

Describe the end behavior of f\left(x\right).

Worked Solution

d

Find where f\left(x\right)=1.

Worked Solution

Consider the piecewise function shown in the graph:

a

Identify the function families in the piecewise function.

Worked Solution

b

Find all zeros and intercepts of the piecewise function.

Worked Solution

Compare the domain, range, and intercepts for each pair of functions.

a

f\left(x\right)=\sqrt{x+2}

Worked Solution

b

f\left(x\right)=-\sqrt{x-1}-2

g\left(x\right)=\sqrt[3]{x}+2

Worked Solution

Idea summary

Graph of y=\sqrt{x}

Graph of y=\sqrt[3]{x}

The graphs of the square root and cube root parent functions are similar for x>0, but the domain of the square root function does not include negative values and the domain of the cube root function does.

Radical functions can be transformed in the following ways:

Square root | Cube root | |
---|---|---|

\text{Parent function:} | y= \sqrt{x} | y= \sqrt[3]{x} |

\text{Reflection across the }x\text{-axis:} | y=-\sqrt{x} | y=-\sqrt[3]{x} |

\text{Reflection across the }y\text{-axis:} | y=\sqrt{-x} | y=\sqrt[3]{-x} |

\text{Vertical stretch when } \left|a\right|>1 \\ \text{Vertical compression when } 0<\left|a\right|<1 \text{:} | y=a\sqrt{x} | y=a\sqrt[3]{x} |

\text{Horziontal compression when } \left|b\right|>1 \\ \text{Horizontal stretch when } 0<\left|b\right|<1 \text{:} | y=\sqrt{bx} | y=\sqrt[3]{bx} |

\text{Horizontal translation by } h \\ \text{Vertical translation by } k \text{:} | y=\sqrt{x-h} + k | y=\sqrt[3]{x-h} + k |

(h, k) \text{:} | \text{Endpoint} | \text{Point of inflection} |

The domain and range of the square root function will change with a reflection, or as h or k changes, while the domain and range of the cube root function will continue to be all real numbers.

The absolute extremum of square root function will change location when translated. If there are no reflections, the endpoint of the domain is an absolute minimum. If a vertical reflection occurs, it becomes an absolute maximum.

Square root functions do not have a relative extremum, and cube root functions have neither absolute nor relative extrema.