Are the following functions inverses? $$ f(x)=\frac{4}{x-4}-5 $$ $$ g(x)=\frac{4}{x+5}+4 $$ No, they are not inverses Yes, they are inverse

Question

UpStudy AI · Accepted Answer

To determine if the functions $$ f(x) $$ and $$ g(x) $$ are inverses, we need to check if $$ f(g(x)) = x $$ and $$ g(f(x)) = x $$.

Let's start by calculating $$ f(g(x)) $$:

1. Substitute $$ g(x) $$ into $$ f(x) $$:
   $$
   f(g(x)) = f\left(\frac{4}{x+5}+4\right)
   $$
   Now, replace $$ x $$ in $$ f(x) $$:
   $$
   f\left(\frac{4}{x+5}+4\right) = \frac{4}{\left(\frac{4}{x+5}+4\right) - 4} - 5
   $$

2. Simplify the expression inside $$ f $$:
   $$
   \frac{4}{x+5} + 4 - 4 = \frac{4}{x+5}
   $$
   So, we have:
   $$
   f(g(x)) = \frac{4}{\frac{4}{x+5}} - 5
   $$

3. Simplifying further:
   $$
   f(g(x)) = (x+5) - 5 = x
   $$

Now, let's calculate $$ g(f(x)) $$:

1. Substitute $$ f(x) $$ into $$ g(x) $$:
   $$
   g(f(x)) = g\left(\frac{4}{x-4}-5\right)
   $$
   Now, replace $$ x $$ in $$ g(x) $$:
   $$
   g\left(\frac{4}{x-4}-5\right) = \frac{4}{\left(\frac{4}{x-4}-5\right) + 5} + 4
   $$

2. Simplify the expression inside $$ g $$:
   $$
   \frac{4}{x-4} - 5 + 5 = \frac{4}{x-4}
   $$
   So, we have:
   $$
   g(f(x)) = \frac{4}{\frac{4}{x-4}} + 4
   $$

3. Simplifying further:
   $$
   g(f(x)) = (x-4) + 4 = x
   $$

Since both $$ f(g(x)) = x $$ and $$ g(f(x)) = x $$, we conclude that the functions $$ f(x) $$ and $$ g(x) $$ are indeed inverses of each other.

Thus, the answer is:
**Yes, they are inverses.**
Yes, they are inverses.

Are the following functions inverses? \( f(x)=\frac{4}{x-4}-5 \) \( g(x)=\frac{4}{x+5}+4 \) No, they are not inverses Yes, they are inverse

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions