Answer:
(-∞, 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, ∞).
The domain of a function is the set of input values, the function can take
The domain of [tex]\mathbf{f \circ g}[/tex] as a union of interval is: [tex]\mathbf{(-\infty, \frac 13)\ u\ (\frac 13, 2)\ u\ (2, \infty)}[/tex]
The functions are given as:
[tex]\mathbf{f(y) = \frac{4}{y - 2}}[/tex]
[tex]\mathbf{g(x) = \frac{5}{3x - 1}}[/tex]
Substitute x for y in f(y)
[tex]\mathbf{f(x) = \frac{4}{x - 2}}[/tex]
The domains of f(x) and g(x) will be the domain of [tex]\mathbf{f \circ g}[/tex]
Set the denominator of [tex]\mathbf{f(x) = \frac{4}{x - 2}}[/tex] to 0
[tex]\mathbf{x - 2 = 0}[/tex]
Solve for x
[tex]\mathbf{x = 2}[/tex]
So, the domain of f(x) is:
[tex]\mathbf{x \ne 2}[/tex]
Set the denominator of [tex]\mathbf{g(x) = \frac{5}{3x - 1}}[/tex] to 0
[tex]\mathbf{3x - 1 = 0}[/tex]
Solve for x
[tex]\mathbf{3x = 1}[/tex]
[tex]\mathbf{x = \frac 13}[/tex]
So, the domain of g(x) is:
[tex]\mathbf{x \ne \frac 13}[/tex]
At this stage, we have:
[tex]\mathbf{x \ne \frac 13}[/tex] and [tex]\mathbf{x \ne 2}[/tex]
This means that:
[tex]\mathbf{f \circ g}[/tex] can take
So, the domain of [tex]\mathbf{f \circ g}[/tex] as a union of interval is:
[tex]\mathbf{(-\infty, \frac 13)\ u\ (\frac 13, 2)\ u\ (2, \infty)}[/tex]
Read more about function domains at:
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