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Chapter 1: Problem 61

Suppose \(f\) and \(g\) are functions. Show that the composition \(f \circ g\) has the same domain as \(g\) if and only if the range of \(g\) is contained in the domain of \(f\).

Short Answer

Expert verified
In conclusion, we showed that if the domain of $f \circ g$ is equal to the domain of $g$, then the range of $g$ is contained within the domain of $f$. We also showed that if the range of $g$ is contained within the domain of $f$, then the domain of $f \circ g$ is equal to the domain of $g$. Therefore, the composition $f \circ g$ has the same domain as $g$ if and only if the range of $g$ is contained in the domain of $f$.

Step by step solution

01

Direction 1: If f 鈭� g has the same domain as g, then the range of g is contained in the domain of f.

Suppose that the domain of f 鈭� g is equal to the domain of g. We want to show that the range of g is contained in the domain of f. Let's recall the definition of the composition of two functions: \[ (f\circ g)(x) = f(g(x)) \] For any element x in the domain of g which is also the domain of f 鈭� g, we can apply g to x (since x is in the domain of g), then apply f to g(x) (since x is also in the domain of f 鈭� g). Thus, g(x) must be a valid input for the function f. This means that g(x) is in the domain of f for any x in the domain of f 鈭� g. Now, let's consider the range of g. The range of g consists of all the possible outputs of g. If we plug an input x from the domain of g into f 鈭� g, we get f(g(x)). Since g(x) is in the domain of f, the range of g must be a subset of the domain of f.
02

Direction 2: If the range of g is contained in the domain of f, then f 鈭� g has the same domain as g.

Now suppose that the range of g is contained in the domain of f. We want to show that f 鈭� g has the same domain as g. Let's consider an element x in the domain of g. Since x is in the domain of g, we can apply g to x and obtain the output g(x). By our assumption, the range of g is contained in the domain of f, so g(x) must be a valid input for the function f. This means that the composition (f 鈭� g)(x) = f(g(x)) is well-defined for any x in the domain of g. Thus, the domain of f 鈭� g consists of all the inputs of g for which f(g(x)) is defined. Since we've shown that this is true for any x in the domain of g, we conclude that the domain of f 鈭� g is equal to the domain of g. In conclusion, we have shown that f 鈭� g has the same domain as g if and only if the range of g is contained in the domain of f.

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