Question

Are these two functions inverse determine algebraicly whether f(x)=3x-2/5x and g(x)=-2/5x-3 are inverse functions

Answer 1

To determine whether the functions \( f(x) = \frac{3x - 2}{5x} \) and \( g(x) = -\frac{2}{5}x - 3 \) are inverse functions, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).

### Let's calculate \( f(g(x)) \):

Substituting \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(-\frac{2}{5}x - 3\right) = \frac{3\left(-\frac{2}{5}x - 3\right) - 2}{5\left(-\frac{2}{5}x - 3\right)}
\]

Now, compute the numerator:
\[
3\left(-\frac{2}{5}x - 3\right) - 2 = -\frac{6}{5}x - 9 - 2 = -\frac{6}{5}x - 11
\]

Then, compute the denominator:
\[
5\left(-\frac{2}{5}x - 3\right) = -2x - 15
\]

Thus,
\[
f(g(x)) = \frac{-\frac{6}{5}x - 11}{-2x - 15}
\]

### Now let's check \( g(f(x)) \):

Substituting \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g\left(\frac{3x - 2}{5x}\right) = -\frac{2}{5}\left(\frac{3x - 2}{5x}\right) - 3
\]

This simplifies to:
\[
-\frac{2(3x - 2)}{25x} - 3 = -\frac{6x - 4}{25x} - 3
\]

To combine the terms:
\[
-3 = -\frac{75x}{25x}
\]

So:
\[
g(f(x)) = -\frac{6x - 4}{25x} - \frac{75x}{25x} = -\frac{6x - 4 + 75x}{25x} = -\frac{81x - 4}{25x}
\]

### Check if \( f(g(x)) = x \) and \( g(f(x)) = x \):

Now we need to simplify or directly check whether \( f(g(x)) \) and \( g(f(x)) \) equal \( x \).

After careful calculations, if we find that neither composition yields \( x \), then \( f \) and \( g \) are not inverses.

### Conclusion:

To fully verify these two functions are inverses algebraically following both compositions through substitution yields complicated expressions that ultimately should return to \( x \) if they are indeed inverses.

Given the initial evaluations and complexities, it appears that \( f(g(x)) \neq x \) and \( g(f(x)) \neq x \).

Therefore, the functions \( f(x) = \frac{3x - 2}{5x} \) and \( g(x) = -\frac{2}{5}x - 3 \) are not inverse functions of each other.