Lesson Objectives

- Learn how to identify the graph of the identity function
- Learn how to identify the graph of the squaring function
- Learn how to identify the graph of the cubing function
- Learn how to identify the graph of the square root function
- Learn how to identify the graph of the cube root function
- Learn how to identify the graph of the reciprocal function

## How to Identify the Graph of a Parent Function

In this lesson, we will show the graphs of six basic functions, which are also known as parent functions. These functions come up often in our study of algebra and math in general. It is a good idea to know the shape, along with the domain and range for these functions. This knowledge will become very useful when we start looking at graphing transformations in a few lessons. Additionally, as we continue to advance through this section and the course, we'll learn about additional parent functions and their corresponding graphs.

### Continuity

We won't get a formal definition of continuity until we reach a Calculus I course. At this point, we can give an informal definition and just say that a function is continuous over an interval of its domain if the graph over that interval can be drawn without lifting the pencil from the paper. Notice in the image above, the function has a discontinuity at the point where x equals 1. If we wanted to draw this graph, we would need to stop where x is 1, lift the pencil up, and place it back down on the paper to continue our drawing. All the functions we will look at in this lesson are continuous over their entire domain except for the reciprocal function.### Identity Function

The identity function pairs every real number with itself. $$f(x)=x$$ $$\text{domain}:(-\infty, \infty)$$ $$\text{range}: (-\infty, \infty)$$ This function is continuous over its entire domain.x | y |
---|---|

-6 | -6 |

0 | 0 |

6 | 6 |

### Squaring Function

The squaring function pairs every real number with its square. $$f(x)=x^2$$ $$\text{domain}:(-\infty, \infty)$$ $$\text{range}: [0, \infty)$$ This function is continuous over its entire domain.x | y |
---|---|

-3 | 9 |

-2 | 4 |

-1 | 1 |

0 | 0 |

1 | 1 |

2 | 4 |

3 | 9 |

### Cubing Function

The cubing function pairs every real number with its cube. $$f(x)=x^3$$ $$\text{domain}:(-\infty, \infty)$$ $$\text{range}: (-\infty, \infty)$$ This function is continuous over its entire domain.x | y |
---|---|

-2 | -8 |

-1 | -1 |

0 | 0 |

1 | 1 |

2 | 8 |

### Square Root Function

The square root function pairs every non-negative real number with its principal square root. Recall that in the real number system, we can't take the square root of a negative number. $$f(x)=\sqrt{x}$$ $$\text{domain}:[0, \infty)$$ $$\text{range}: [0, \infty)$$ This function is continuous over its entire domain.x | y |
---|---|

0 | 0 |

1 | 1 |

4 | 2 |

9 | 3 |

### Cube Root Function

The cube root function pairs every real number with its cube root. $$f(x)=\sqrt[3]{x}$$ $$\text{domain}:(-\infty, \infty)$$ $$\text{range}: (-\infty, \infty)$$ This function is continuous over its entire domain.x | y |
---|---|

-8 | -2 |

0 | 0 |

8 | 2 |

### Reciprocal Function

The reciprocal function pairs every non-zero real number with its reciprocal. Since x is in the denominator, x can't be zero (division by zero is undefined). $$f(x)=\frac{1}{x}$$ $$\text{domain}:(-\infty, 0) ∪ (0, \infty)$$ $$\text{range}:(-\infty, 0) ∪ (0, \infty)$$ This function is discontinuous at x = 0.x | y |
---|---|

-2 | -1/2 |

-1 | -1 |

-1/2 | -2 |

1/2 | 2 |

1 | 1 |

2 | 1/2 |

#### Skills Check:

Example #1

Which parent function pairs every real number with itself?

Please choose the best answer.

A

$$f(x)=x$$

B

$$f(x)=x^2$$

C

$$f(x)=\sqrt{x}$$

D

$$f(x)=x^3$$

E

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

Example #2

Which x-value(s) is(are) excluded from the domain of the reciprocal function?

Please choose the best answer.

A

$$0$$

B

$$x < 0$$

C

$$x > 0$$

D

None

E

All

Example #3

Determine if the following is true or false.

The cube root function has a range from 0 (inclusive) to positive infinity. $$\text{Range}:[0, \infty)$$

Please choose the best answer.

A

True

B

False

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