Question

How do you find the domain and range of `1/(x+6)`?

Answer 1

Domain: `" "x in RR, x ne 6`

Range: `" "y in RR, y ne 0`

Explanation:

The domain is all possible values of `x` for which `1/(x+6)` is defined.

We see that the function is only undefined if the denominator is 0, meaning that `x+6 = 0`

This tells us that `x` cannot be `-6`.

So we can say our domain is: `" "x in RR, x ne -6`

(This is just a fancy way of saying "`x` can be all real numbers except for -6")

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Range is a little harder to find. We need to find all possible values that `1/(x+6)` could be.

Let's think about it this way: what does the graph of `y = 1/(x+6)` look like? It will be the graph of `y = 1/x`, but translated 6 units to the left, like this:
graph{y = 1/(x+6) [-13.71, 6.29, -4.76, 5.24]}

We need to find all possible `y` values that can be generated from this graph.

When `x < -6`, we can see that the function `1/(x+6)` will be negative, since `x+6` will be negative.

As we approach `x = -6` from the left side, the function flies downwards towards `-oo`, hitting every possible negative value.

As we approach `x = -oo`, the function tends towards zero, but never actually reaches it. This is because the denominator is getting bigger and bigger, so the fraction is getting closer and closer to 0 without ever reaching it.

Therefore, from `x = -oo` to `x=-6`, we can say that we will hit all possible negative values of `y`.

The same logic can be used for the positive side of the graph.

As we approach `x = -6` from the right side, the function flies upwards towards `oo`, hitting every possible positive value.

As we approach `x = oo`, the function tends towards zero but never actually reaches it.

Therefore, from `x = -6` to `x = oo`, we can say that we will hit all possible positive values of `y`.

We've checked every possible `x` value now, and our possible `y` values are:

`y` is a real number
`y` is positive OR negative

In other words:

`y in RR, y ne 0`

This is the range of `1/(x+6`

Final Answer