Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

Given the functions below, find the domain of [tex]\( (f \circ g)(x) \)[/tex].

[tex]\( f(x) = \frac{1}{-10x-7} \)[/tex]
[tex]\( g(x) = \frac{1}{-6x+5} \)[/tex]

Select the correct answer below:
A. All real numbers except [tex]\( \frac{5}{6} \)[/tex] and [tex]\( \frac{15}{14} \)[/tex]
B. All real numbers except [tex]\( \frac{5}{6} \)[/tex] and [tex]\( \frac{47}{42} \)[/tex]
C. All real numbers except [tex]\( \frac{5}{6} \)[/tex] and [tex]\( \frac{41}{42} \)[/tex]
D. All real numbers except [tex]\( \frac{5}{6} \)[/tex] and [tex]\( \frac{13}{14} \)[/tex]
E. All real numbers except [tex]\( \frac{5}{6} \)[/tex] and [tex]\( \frac{53}{42} \)[/tex]
F. All real numbers except [tex]\( \frac{5}{6} \)[/tex] and [tex]\( -\frac{7}{10} \)[/tex]

Sagot :

To find the domain of the composite function [tex]\((f \circ g)(x)\)[/tex], we need to determine where both functions are defined and where the composite function itself is defined. Let's break this problem into steps.

1. Identify the domain of [tex]\(g(x)\)[/tex]:

The function [tex]\(g(x) = \frac{1}{-6x + 5}\)[/tex] is defined for all [tex]\(x\)[/tex] except where the denominator is zero. Set the denominator equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ -6x + 5 = 0 \implies x = \frac{5}{6} \][/tex]

Therefore, [tex]\(g(x)\)[/tex] is defined for all [tex]\(x \neq \frac{5}{6}\)[/tex].

2. Identify the domain of [tex]\(f(g(x))\)[/tex]:

The function [tex]\(f(x) = \frac{1}{-10x - 7}\)[/tex] requires that [tex]\(x \neq -\frac{7}{10}\)[/tex] to be defined. Therefore, [tex]\(f\)[/tex] is defined for all [tex]\(x \neq -\frac{7}{10}\)[/tex]. In our case, we need this to [tex]\(f(g(x))\)[/tex].

To find where [tex]\(f(g(x))\)[/tex] is defined, we substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f\left( \frac{1}{-6x + 5} \right) = \frac{1}{-10 \left( \frac{1}{-6x + 5} \right) - 7} \][/tex]

3. Determine when [tex]\(f(g(x))\)[/tex] is undefined:

The function [tex]\(f(g(x))\)[/tex] becomes undefined when its denominator is zero:
[tex]\[ -10 \left( \frac{1}{-6x + 5} \right) - 7 = 0 \][/tex]
Simplify inside the denominator:
[tex]\[ - \frac{10}{-6x + 5} - 7 = 0 \][/tex]
Rearrange to solve for [tex]\(x\)[/tex]:
[tex]\[ - \frac{10}{-6x + 5} = 7 \implies \frac{10}{6x - 5} = 7 \implies 10 = 7(6x - 5) \implies 10 = 42x - 35 \implies 42x = 45 \implies x = \frac{45}{42} = \frac{15}{14} \][/tex]

Therefore, the domain of the composite function [tex]\((f \circ g)(x)\)[/tex] is all real numbers except where the original [tex]\(g(x)\)[/tex] or the intermediate steps of [tex]\(f(g(x))\)[/tex] become undefined.

Given the calculations:
- [tex]\(g(x)\)[/tex] is undefined at [tex]\(x = \frac{5}{6}}. - \(f(g(x))\)[/tex] is undefined at [tex]\(x = \frac{15}{14}}. Thus, the domain of \((f \circ g)(x)\)[/tex] is all real numbers except [tex]\(x = \frac{5}{6}\)[/tex] and [tex]\(x = \frac{15}{14}\)[/tex].

The correct answer is:

All real numbers except [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{15}{14} \)[/tex].

Therefore, the correct option is:
#### All real numbers except [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{15}{14}\)[/tex].
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.