Find f​(g​(x​)) and g​(f​(x​)) and determine whether the pair of functions f and g are inverses of each other.

f(x)=4x-5 and g(x)=(x+4)/5

To determine whether \( f(x) = 4x - 5 \) and \( g(x) = \frac{x + 4}{5} \) are inverses of each other, we need to compute two compositions:

1. \( f(g(x)) \)
2. \( g(f(x)) \)

If both compositions simplify to \( x \), then \( f(x) \) and \( g(x) \) are inverses of each other.

### Step 1: Find \( f(g(x)) \)
Start by plugging \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f\left( \frac{x + 4}{5} \right)
\]
Substitute \( \frac{x + 4}{5} \) into the function \( f(x) = 4x - 5 \):

\[
f\left( \frac{x + 4}{5} \right) = 4 \times \left( \frac{x + 4}{5} \right) - 5
\]
Simplify the expression:

\[
= \frac{4(x + 4)}{5} - 5
= \frac{4x + 16}{5} - 5
\]
Now, rewrite \( 5 \) as \( \frac{25}{5} \) to subtract:

\[
f(g(x)) = \frac{4x + 16 - 25}{5} = \frac{4x - 9}{5}
\]

So, \( f(g(x)) = \frac{4x - 9}{5} \), which does not simplify to \( x \). Therefore, \( f(g(x)) \neq x \).

### Step 2: Find \( g(f(x)) \)
Now, plug \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(4x - 5)
\]
Substitute \( 4x - 5 \) into the function \( g(x) = \frac{x + 4}{5} \):

\[
g(4x - 5) = \frac{(4x - 5) + 4}{5}
\]
Simplify the expression:

\[
= \frac{4x - 5 + 4}{5} = \frac{4x - 1}{5}
\]

So, \( g(f(x)) = \frac{4x - 1}{5} \), which does not simplify to \( x \). Therefore, \( g(f(x)) \neq x \).

### Conclusion:
Since neither \( f(g(x)) \) nor \( g(f(x)) \) simplifies to \( x \), the functions \( f(x) = 4x - 5 \) and \( g(x) = \frac{x + 4}{5} \) are **not inverses** of each other.

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Would you like any further explanation or details?

Here are 5 related questions to expand your understanding:
1. How do we determine if two functions are inverses of each other in general?
2. What would \( f(g(x)) \) and \( g(f(x)) \) look like if the functions were inverses?
3. Can the domain and range of inverse functions affect their properties?
4. How does graphing functions and their inverses help in understanding their relationship?
5. What are other methods to find the inverse of a function algebraically?

**Tip**: Always check the domain of the functions when dealing with inverses, as domain restrictions can affect the outcome.

Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other.
f(x)=4x-5 and g(x)=(x+4)/5

Find f(g(x)) and g(f(x)) to determine if the functions f(x) = 4x - 5 and g(x) = (x + 4)/5 are inverses of each other. Detailed explanation with key algebraic concepts like function composition and the inverse function theorem. Suitable for Grades 10-12.

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