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QuestionFind (f∘g)(x)(f \circ g)(x) and its domain in interval notation for f(x)=xx−1f(x)=\frac{x}{x-1} and g(x)=13x2−36g(x)=\frac{13}{x^{2}-36}.

Studdy Solution

STEP 1

Assumptions1. The functions are given as f(x)=xx−1f(x)=\frac{x}{x-1} and g(x)=13x−36g(x)=\frac{13}{x^{}-36}. . We are asked to evaluate the composite function (f∘g)(x)(f \circ g)(x) and find its domain in interval notation.

STEP 2

The composite function (f∘g)(x)(f \circ g)(x) is defined as f(g(x))f(g(x)). So, we need to replace xx in f(x)f(x) with g(x)g(x).
f(g(x))=f(13x2−36)f(g(x)) = f\left(\frac{13}{x^{2}-36}\right)

STEP 3

Substitute g(x)g(x) into f(x)f(x).
f(g(x))=13x2−3613x2−36−1f(g(x)) = \frac{\frac{13}{x^{2}-36}}{\frac{13}{x^{2}-36}-1}

STEP 4

implify the denominator by finding a common denominator.
f(g(x))=13x2−3613−x2+36x2−36f(g(x)) = \frac{\frac{13}{x^{2}-36}}{\frac{13-x^{2}+36}{x^{2}-36}}

STEP 5

implify the fraction.
f(g(x))=1349−x2f(g(x)) = \frac{13}{49-x^{2}}

STEP 6

Now, we need to find the domain of (f∘g)(x)(f \circ g)(x). The domain of a function is the set of all real numbers for which the function is defined. In this case, we need to find all xx such that 49−x2≠049-x^{2} \neq0 and x2−36≠0x^{2}-36 \neq0.

STEP 7

olve the equation 49−x2≠049-x^{2} \neq0.
49−x2≠0⇒x2≠49⇒x≠±749-x^{2} \neq0 \Rightarrow x^{2} \neq49 \Rightarrow x \neq \pm7

STEP 8

olve the equation x2−36≠0x^{2}-36 \neq0.
x2−36≠0⇒x2≠36⇒x≠±6x^{2}-36 \neq0 \Rightarrow x^{2} \neq36 \Rightarrow x \neq \pm6

STEP 9

Combine the solutions to find the domain of (f∘g)(x)(f \circ g)(x).
The domain of (f∘g)(x)(f \circ g)(x) is all real numbers except −7,−6,6,7-7, -6,6,7.

STEP 10

Write the domain in interval notation.
The domain of (f∘g)(x)(f \circ g)(x) in interval notation is (−∞,−7)∪(−7,−6)∪(−6,6)∪(6,7)∪(7,∞)(-\infty, -7) \cup (-7, -6) \cup (-6,6) \cup (6,7) \cup (7, \infty).
So, the composite function (f∘g)(x)=1349−x2(f \circ g)(x)=\frac{13}{49-x^{2}} and its domain is (−∞,−7)∪(−7,−6)∪(−6,6)∪(6,7)∪(7,∞)(-\infty, -7) \cup (-7, -6) \cup (-6,6) \cup (6,7) \cup (7, \infty).

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