# For the given functions f and g, find `f@g` and state the domain of `f@g`.`f(x)= lnx` `g(x)=|x| -7`

txmedteach | Certified Educator

For this, you can take two approaches. Our first approach will look at how the domains and ranges of each function line up. Let's start by establishing the domains and ranges of each individual function.

For` f(x)`:

Domain: `x in RR^+`

Range: `y in RR`

For `g(x)`:

Domain: `x in RR`

Range: `y in RR, y > -7`

We know the range for `g(x)` based on the minimum value of `|x|` being 0, giving the minimum value in the range to be -7.

Now that we have established the domain and range, let's look at how a composition of functions lines up. The range of `g(x)` must be exactly the domain of `f(x)` in order to allow each possible `x` have a defined output for `f@g`. This face is due to the following interpretation of a composition of functions:

`f@g = f(g(x))`

Here, you can see that `g(x)` is the "input" to the function `f(x)`. Therefore, the output of `g(x)` has the same limit as the input to `f(x)`. This fact is why the range of `g(x)` must be the domain of `f(x)`.

So, we need to set the domain of `g(x)` so that the range can only be the positive real numbers. In other words, over the domain of the composition of functions,

`g(x) > 0`

So we need solve for the x-values that will limit our domain such that this condition is met.

`|x|-7 > 0`

Simplifying by adding 7 to both sides:

`|x| > 7`

Now, we have two resulting possibilities, if `x` is positive, then we can just get rid of the absolute value bars. If `x` is negative, we are actually taking the negative value of `x` to get the positive result.

`x > 7` or `-x>7`

The relation on the left is pretty solid. However, to solve the relation on the right, we need to multiply by -1 on both sides. However, when you multiply an inequality by -1, you actually must flip the inequality sign. This gives us the domain for x:

`x > 7` or `x < -7`

There's your answer! Now, let's solve it a different way. This other way is effectively the same, but it may be conceptually easier to understand.

Let's consider what `f@g` will be. We basically substitute the `x` term in `f(x)` with the function `g(x)`:

`f@g = ln(g(x)) = ln(|x|-7)`

Now, we know that the natural logarithm is only defined for values greater than zero. Therefore, the domain will be found by setting the term inside the logarithm greater than zero:

`|x|-7>0`

So, we end up exactly where we ended up in the previous method.