What is the domain of the function f \circ g where f(y)= \frac{4}{y-2} and g(x)= \frac{5}{3x-1} ? Express it in terms of a union of intervals of real numbers.

The domain of a function is the set of input values, the function can take

The domain of as a union of interval is:

The functions are given as:

Substitute x for y in f(y)

The domains of f(x) and g(x) will be the domain of

Set the denominator of to 0

Solve for x

So, the domain of f(x) is:

Set the denominator of to 0

Solve for x

So, the domain of g(x) is:

At this stage, we have:

and

This means that:

can take

So, the domain of as a union of interval is:

Read more about function domains at:

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(-∞, ⅓) ⋃ (⅓, 2) ⋃ (2, ∞)

Step-by-step explanation:

f(x) = 4 / (x − 2)

g(x) = 5 / (3x − 1)

The domain of f(x) is x ≠ 2, or (-∞, 2) ⋃ (2, ∞).

The domain of g(x) is x ≠ ⅓, or (-∞, ⅓) ⋃ (⅓, ∞).

Therefore, the domain of f(g(x)) is the overlap:

x ≠ ⅓ or 2, or (-∞, ⅓) ⋃ (⅓, 2) ⋃ (2, ∞).

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