Question 1186928
Let's analyze each statement:

**a. All elements in the domain of *g* must be in the domain of *f o g***

**True.**

*Explanation:* The composition *f o g* (or *f(g(x))*) is defined as applying the function *g* first, and then applying the function *f* to the result.  For *f(g(x))* to be defined, *g(x)* *must* be defined.  This means that *x* must be in the domain of *g*.  If *x* is not in the domain of *g*, then *g(x)* is undefined, and therefore *f(g(x))* is also undefined.  So, any *x* in the domain of *f o g* *must* also be in the domain of *g*.

**b. All elements in the range of *f* must be in the range of *f o g***

**False.**

*Counterexample:*

Let *f(x) = x²*.  The range of *f* is [0, ∞).

Let *g(x) = x + 1*.

Then *(f o g)(x) = f(g(x)) = f(x + 1) = (x + 1)²*. The range of *f o g* is also [0, ∞).

However, let's consider a slightly different g(x)

Let *f(x) = x²*.  The range of *f* is [0, ∞).

Let *g(x) = 0*.

Then *(f o g)(x) = f(g(x)) = f(0) = 0²=0*. The range of *f o g* is just {0}.

In this case, the range of *f* is [0, ∞), but the range of *f o g* is only {0}.  Therefore, not all elements in the range of *f* are in the range of *f o g*.

In general, the range of *f o g* is a *subset* of the range of *f*. It can be smaller, but not larger.