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:

link

(-∞, 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, ∞).

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for