Question 1186928: For each of the following statements, explain why it is true for all functions f and g or an example of specific functions f and g showing that it is false.
a. All elements in the domain of g must be in the domain of f o g (fog)
b. All elements in the range of f must be in the range of f o g
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! 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.
Answer by ikleyn(52835)  (Show Source):
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