To find [tex]\((f \circ g)(x)\)[/tex], we first need to understand how function composition works. The notation [tex]\((f \circ g)(x)\)[/tex] means we first apply [tex]\(g(x)\)[/tex] and then apply [tex]\(f\)[/tex] to the result of [tex]\(g(x)\)[/tex].
Let's start with defining the functions:
[tex]\[ f(x) = \frac{1}{x^2 - 9} \][/tex]
[tex]\[ g(x) = \frac{1}{x} \][/tex]
### Step 1: Calculate [tex]\(g(x)\)[/tex]
The function [tex]\(g(x)\)[/tex] is straightforward:
[tex]\[ g(x) = \frac{1}{x} \][/tex]
### Step 2: Calculate [tex]\(f(g(x))\)[/tex]
Next, we need to find [tex]\(f(g(x))\)[/tex], which means we substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{1}{x}\right) \][/tex]
### Step 3: Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]
We substitute [tex]\(\frac{1}{x}\)[/tex] for [tex]\(x\)[/tex] in the function [tex]\(f\)[/tex]:
[tex]\[ f\left(\frac{1}{x}\right) = \frac{1}{\left(\frac{1}{x}\right)^2 - 9} \][/tex]
Now simplify the expression inside the denominator:
[tex]\[ \left(\frac{1}{x}\right)^2 = \frac{1}{x^2} \][/tex]
Substitute back:
[tex]\[ f\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x^2} - 9} \][/tex]
To simplify further, find a common denominator within the fraction:
[tex]\[ \frac{1}{\frac{1 - 9x^2}{x^2}} \][/tex]
Invert the fraction in the denominator:
[tex]\[ f\left(\frac{1}{x}\right) = \frac{x^2}{1 - 9x^2} \][/tex]
So,
[tex]\[ (f \circ g)(x) = \frac{x^2}{1 - 9x^2} \][/tex]
### Step 4: Determine the Domain of [tex]\((f \circ g)(x)\)[/tex]
We need to identify values of [tex]\(x\)[/tex] that make [tex]\((f \circ g)(x)\)[/tex] undefined.
1. [tex]\(g(x) = \frac{1}{x}\)[/tex] is undefined when [tex]\(x = 0\)[/tex].
2. The denominator of [tex]\(f(g(x))\)[/tex] must not be equal to zero:
[tex]\[ \frac{1}{x^2} - 9 \neq 0 \][/tex]
[tex]\[ \frac{1}{x^2} \neq 9 \][/tex]
[tex]\[ x^2 \neq \frac{1}{9} \][/tex]
[tex]\[ x \neq \pm \frac{1}{3} \][/tex]
Combining these conditions, [tex]\(x\)[/tex] must satisfy:
[tex]\[ x \neq 0, \pm \frac{1}{3} \][/tex]
Therefore, the domain of [tex]\((f \circ g)(x)\)[/tex] is all real numbers except [tex]\(0, \pm \frac{1}{3}\)[/tex].
Conclusion:
[tex]\[ (f \circ g)(x) = \frac{x^2}{1 - 9x^2} \][/tex]
Domain: [tex]\( x \neq 0, \pm \frac{1}{3} \)[/tex]
