Questions: QUESTION 5 Determine whether f(x) and g(x) are inverses of each other. f(x)=3/2 x+6 g(x)=2/3 x-4 There is not enough information to solve No, they are not inverses Yes, they are inverses

QUESTION 5 Determine whether f(x) and g(x) are inverses of each other. f(x)=3/2 x+6 g(x)=2/3 x-4 There is not enough information to solve No, they are not inverses Yes, they are inverses

Transcript text: QUESTION 5 Determine whether $f(x)$ and $g(x)$ are inverses of each other. \[ \begin{array}{l} f(x)=\frac{3}{2} x+6 \\ g(x)=\frac{2}{3} x-4 \end{array} \] There is not enough information to solve No, they are not inverses Yes, they are inverses

Solution

Solution Steps

To determine if two functions $ f(x) $ and $ g(x) $ are inverses of each other, we need to check if $ f(g(x)) = x $ and $ g(f(x)) = x $. If both conditions are satisfied, then the functions are inverses.

Step 1: Evaluate $ f(g(x)) $

We start by substituting $ g(x) $ into $ f(x) $: \[ f(g(x)) = f\left(\frac{2}{3}x - 4\right) = \frac{3}{2}\left(\frac{2}{3}x - 4\right) + 6 \] Calculating this gives: \[ f(g(x)) = \frac{3}{2} \cdot \frac{2}{3}x - 6 + 6 = 1.0x \] Thus, $ f(g(x)) = x $.

Step 2: Evaluate $ g(f(x)) $

Next, we substitute $ f(x) $ into $ g(x) $: \[ g(f(x)) = g\left(\frac{3}{2}x + 6\right) = \frac{2}{3}\left(\frac{3}{2}x + 6\right) - 4 \] Calculating this gives: \[ g(f(x)) = \frac{2}{3} \cdot \frac{3}{2}x + 4 - 4 = 1.0x \] Thus, $ g(f(x)) = x $.

Step 3: Conclusion

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