What does this, {x|x __<__ 0}, mean?

My question is, "What is the domain and range of this function: y = 1/x ?"

One of the options is Domain: {x|x __<__ 0}, but I don't know what it means to tell if it's right or not.

SurpriseMe
Jul 3, 2017

#1**+1 **

Domain and range of the function \(y=\frac{1}{x}\)

Now, let's figure out the domain first. The domain of a function just means the possible range of x-values that output a real y-value.

Let's consider that. Is there any number for *x *that would output a nonreal y-value?

Yes, 0. If you plug in 0 for *x, *you get a nonreal y-value. Will any others? No. I've made a table of the first few domain options.

Possible Domain | Corresponding y-value | Is it real? |

-5 | \(\frac{1}{-5}\) | Yes |

-4 | \(\frac{1}{-4}\) | Yes |

-3 | \(\frac{1}{-3}\) | Yes |

-2 | \(\frac{1}{-2}\) | Yes |

-1 | \(\frac{1}{-1}\) | Yes |

0 | \(\frac{1}{0}\) | No, a number divided by 0 is undefined. |

1 | \(\frac{1}{1}\) | Yes |

2 | \(\frac{1}{2}\) | Yes |

3 | \(\frac{1}{3}\) | Yes |

4 | \(\frac{1}{4}\) | Yes |

5 | \(\frac{1}{5}\) | Yes |

You can see that any numer, except 0, will output a real y-value. Therefore, the domain is any number that isn't 0.

Domain: \(\{x\in {\rm I\!R}\hspace{1mm}: x\neq0\}\)

In other words, the domain* *is apart of the real numbers except 0.

Let's think about the range, too. The range is set of all possible values that the function can actually produce. Let's try making a table for that:

Possible Range | Corresponding x-value | Is it real? |

-3 | \(-\frac{1}{3}\) | Yes |

-2 | \(-\frac{1}{2}\) | Yes |

-1 | \(-1\) | Yes |

0 | No solution | No |

1 | \(1\) | Yes |

2 | \(\frac{1}{2}\) | Yes |

3 | \(\frac{1}{3}\) | Yes |

You might be wondering why 0 has "no solution" on it. Well, let's try and solve it:

\(y=\frac{1}{x}\) | Substitute in 0 for y. |

\(0=\frac{1}{x}\) | Multiply my x on both sides. |

\(0=1\) | This is a nonsensical answer. Since \(0\neq1\), there is no solution. Therefore, 0 cannot be a possible range value. |

Any other real number works, so our range is:

\(\{f(x)\in {\rm I\!R}\hspace{1mm}:f(x)\neq0\}\)

To recap, the domain and range is as follows

Domain: \(\{x\in {\rm I\!R}\hspace{1mm}: x\neq0\}\)

Range: \(\{f(x)\in {\rm I\!R}\hspace{1mm}:f(x)\neq0\}\)

To answer your second question about what does \(\{x|x\leq0\}\) mean. It means that x is a member of the real numbers such that x is nonnegative (0, 2/9, 1, π, 4.83333..., etc.). However, x has to be nonnegative, so it can't be -3/4, -5, -8.980333333..., etc.

TheXSquaredFactor
Jul 3, 2017

#2**+1 **

Thank you for your long response! I do actually know how to do the problem, I just didn't understand what that symbol meant in relation to the question. Thanks though.

SurpriseMe
Jul 3, 2017

#3**+3 **

I think {x|x __<__ 0} is read like this: "x such that x is less than or equal to zero."

You can just read that vertical line like it says, "such that" ...that's how my teacher said it

I finally found a page that agrees with this... here . This might be a helpful page if you have any more questions about symbols like that.

hectictar
Jul 3, 2017