If f(x)=1/x-5 and g(x)=1/x-5
(a) f(g(x))=f(g(x))=
(b) g(f(x))=

Let's go step by step to solve these compositions. 

Given:  
\( f(x) = \frac{1}{x} - 5 \)  
\( g(x) = \frac{1}{x} - 5 \)  

We need to determine:  
(a) \( f(g(x)) \)  
(b) \( g(f(x)) \)  

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### (a) \( f(g(x)) \)
By definition of function composition, \( f(g(x)) \) means substituting \( g(x) \) into \( f(x) \).  

So:  
\[
f(g(x)) = f\left(\frac{1}{x} - 5\right)
\]  

Now substitute \( \frac{1}{x} - 5 \) in place of \( x \) in \( f(x) = \frac{1}{x} - 5 \):  
\[
f\left(\frac{1}{x} - 5\right) = \frac{1}{\frac{1}{x} - 5} - 5
\]

To simplify, let \( u = \frac{1}{x} - 5 \). Then:
\[
\frac{1}{u} = \frac{1}{\frac{1}{x} - 5} = \frac{x}{1 - 5x}
\]

Thus:
\[
f(g(x)) = \frac{x}{1 - 5x} - 5
\]

---

### (b) \( g(f(x)) \)
Similarly, \( g(f(x)) \) means substituting \( f(x) \) into \( g(x) \).  

So:  
\[
g(f(x)) = g\left(\frac{1}{x} - 5\right)
\]  

Now substitute \( \frac{1}{x} - 5 \) in place of \( x \) in \( g(x) = \frac{1}{x} - 5 \):  
\[
g\left(\frac{1}{x} - 5\right) = \frac{1}{\frac{1}{x} - 5} - 5
\]

Following the same steps as above, we get:
\[
g(f(x)) = \frac{x}{1 - 5x} - 5
\]

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### Final Results:
(a) \( f(g(x)) = \frac{x}{1 - 5x} - 5 \)  
(b) \( g(f(x)) = \frac{x}{1 - 5x} - 5 \)

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Do you need further clarifications or a more detailed breakdown of the steps?

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Here are 5 related questions to explore further:
1. What are the domains of \( f(x) \) and \( g(x) \)?
2. How would the compositions change if \( f(x) = \frac{1}{x + 5} \)?
3. What is the inverse function of \( f(x) = \frac{1}{x} - 5 \)?
4. Are \( f \) and \( g \) bijective functions?
5. How do compositions behave when both functions are identical?

**Tip:** When simplifying complex fractions, multiply the numerator and denominator by the least common denominator to avoid mistakes.

If f(x) = 1/x - 5 and g(x) = 1/x - 5, find (a) f(g(x)) and (b) g(f(x))

Discover how to solve function compositions for f(x) = 1/x - 5 and g(x) = 1/x - 5. This guide provides a detailed step-by-step solution to determine f(g(x)) and g(f(x)), covering algebraic principles and function composition concepts. Suitable for students in Grades 10-12.

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