Question

Determine algebraically whether f(x)=
5x
3x−2
​
and g(x)=
5x−3
−2
​
are inverse functions.
Option #1: Yes, the two functions are inverse functions.
Option #2: No, the two functions are not inverse functions.
Option # □ is correct.

Answer 1

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

### Step 1: Compute \( f(g(x)) \)

1. Substitute \( g(x) = 5x - 3 \) into \( f(x) \):
\[
f(g(x)) = f(5x - 3) = \frac{5(5x - 3)}{3(5x - 3) - 2}
\]

2. Simplify the numerator:
\[
5(5x - 3) = 25x - 15
\]

3. Simplify the denominator:
\[
3(5x - 3) - 2 = 15x - 9 - 2 = 15x - 11
\]

4. Therefore:
\[
f(g(x)) = \frac{25x - 15}{15x - 11}
\]

### Step 2: Check if \( f(g(x)) = x \)

Unfortunately, to find if \( \frac{25x - 15}{15x - 11} = x \) simplifies to an identity, we can cross-multiply:
\[
25x - 15 = x(15x - 11)
\]

This expands to:
\[
25x - 15 = 15x^2 - 11x
\]
Rearranging gives us:
\[
15x^2 - 36x + 15 = 0
\]
To check the nature of this quadratic equation, we calculate the discriminant:
\[
D = (-36)^2 - 4(15)(15) = 1296 - 900 = 396
\]
Since \( D > 0 \), it has two distinct real roots. Therefore, \( f(g(x)) \neq x \) for all \( x \).

### Step 3: Compute \( g(f(x)) \)

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

2. Simplifying gives:
\[
5\left(\frac{5x}{3x - 2}\right) = \frac{25x}{3x - 2}
\]
Thus:
\[
g(f(x)) = \frac{25x}{3x - 2} - 3
\]

3. To combine, convert \( -3 \) to a fraction:
\[
-3 = \frac{-9(3x - 2)}{3x - 2} = \frac{-9(3x - 2)}{3x - 2}
\]
Therefore:
\[
g(f(x)) = \frac{25x - 9(3x - 2)}{3x - 2} = \frac{25x - 27x + 18}{3x - 2} = \frac{-2x + 18}{3x - 2}
\]

### Step 4: Check if \( g(f(x)) = x \)

Thus:
\[
g(f(x)) = \frac{-2x + 18}{3x - 2}
\]
To verify if this equals \( x \), we can cross-multiply:
\[
-2x + 18 = x(3x - 2)
\]
Which expands to:
\[
-2x + 18 = 3x^2 - 2x
\]
This simplifies to:
\[
3x^2 - 18 = 0
\]
This is not an identity since it gives specific points (i.e., solutions).

### Conclusion

Since \( f(g(x)) \neq x \) and \( g(f(x)) \neq x \), we conclude:

**Option #2: No, the two functions are not inverse functions.**