Math Lesson 16.4.9 - Range of a Composite Function

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Welcome to our Math lesson on Range of a Composite Function, this is the ninth lesson of our suite of math lessons covering the topic of Composite Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Range of a Composite Function

The range R of a composite function is identified in the same way as for simple functions, i.e. by checking for any restrictions that give not allowed values. However, in composite functions this procedure is carried out after making all operations, that is after obtaining the final form of the function. This procedure is explained more in detail in the example below.

Example 8

Two functions,

are given. Find:

1. The domain of f ◦ g (x)
2. The range of f ◦ g (x)
3. The range of g ◦ f (x)

Solution 8

1. First, we identify the domain of the two individual functions; then we find the domain of f ◦ g by taking the common elements of these two domains, as described in the previous paragraph. Thus, since the argument of a logarithm cannot be zero or negative, we solve the inequality 1 - x > 0 for the domain of the inner function g(x). Hence, we obtain x < 1, i.e. Y = (-∞, 1).
   As for the domain of the outer function f(x), it is clear that all values except x = -2 are allowed, as for this value the denominator becomes zero (it gives a not determined result). Hence, the domain Z of this function includes two intervals: (-∞, -2) and (-2, + ∞). In symbols, we write Z = (-∞, -2) ⋃ (-2, + ∞) where '⋃' is the symbol used to describe the union of two sets, which we are going to discuss in the next chapter.
   In this way, the domain set obtained from the common elements of the two individual domains above is
   R = Y∩Z = (-∞, -2)∪(-2, 1)
   where '⋂' is the symbol used for the intersection (common elements) of two sets.
2. To find the range of f ◦ g (x), we have to identify the function f ◦ g (x) first. Thus,
   f ∘ g(x) = f[g(x)]
   = 3/log⁡(1 - x) + 2
   The range of this composite function consists of all values obtained when making the argument of the logarithm positive. Given this restriction, it is clear that all the resulting values of f ◦ g (x) are positive but smaller than 3/2, because the denominator will always be greater than 2. Hence, we write R(f ◦ g) = (0, 3/2).
3. We use the same procedure as in (b) to find the range of g ◦ f (x). Thus, initially, we identify the function g ◦ f (x). We have
   g ∘ f(x) = g[f(x)]
   = log⁡(1 - 3/x + 2)
   = log⁡(x + 2/x + 2 - 3/x + 2)
   = log⁡(x - 1/x + 2)
   The argument must be positive, so we have to solve the inequality
   x - 1/x + 2 > 0
   The sets of allowed values in the argument correspond to the domain of f ◦ g (x) found in (a). It must be positive, so, we have R = (0, + ∞).

You have reached the end of Math lesson 16.4.9 Range of a Composite Function. There are 9 lessons in this physics tutorial covering Composite Functions, you can access all the lessons from this tutorial below.

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