Question

If `f(x)=x^2` and `g(x)=-12x+7`, how do you find the domain and range of f(x), g(x), and f(g(x))?

Answer 1

Consider the function input values that work or don't work, and consider the possible function output values from all input values. Specific answers given below.

Explanation:

We'll assume we're working with real numbers here i.e. not complex numbers. Ignore that sentence if you don't know what it means; it's not important to your purposes.

`f(x)=x^2` can be applied to any `x`, so its domain is `[-oo,oo]`. Its output is always positive - because squaring turns a number from negative to positive. So its range is `[0,oo]`.

`g(x)=-12x+7` is simpler. It takes all values, and it returns values not limited to any particular range. Both its domain and range are `[-oo,oo]`.

Happily the range of `g` matches the domain of `f`, and so the domain of `f(g(x))` is equal to the domain of `g(x)`,`[-oo,oo]` while its range is equal to the range of `f(x)`, `[0,oo]`.