Question

Given `f(x)=|x|` and `g(x)=5x+1` Find `f(g(x))` and domain and range?

Answer 1

`x=-1/5,1/5`
DOMAIN: `(-∞,+∞)`
RANGE: `(1,+∞)`

Explanation:

`f(g(x))` is substituting `5x+1` into the `x` variable in `f(x) = |x|`

You would start off by doing this:
`f(x) = |5x+1|`

To solve this, you would change the `+` sign into a `-` sign and solve both equations

EQUATION ONE (with `+`):
`0 = 5x+1`
`-1 =5x`
`-1/5 =x`

EQUATION TWO (with `-`):
`0=5x-1`
`1=5x`
`1/5=x`

The domain is the set of all possible x-values and the range is the set of all possible y-values.

So the domain of `f(g(x))` (also known as `f(x) = |5x+1|`) would be
`(-∞,+∞)` or All Real Numbers because the graph starts from negative infinity and proceeds on to positive infinity.

The range would be `(1,+∞)` because the graph starts at `1` on the `y` axis and goes up to infinity (`∞`).