To determine which pair of functions are inverse functions, we need to check if [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] satisfy the following conditions:
1. [tex]\( g(f(x)) = x \)[/tex]
2. [tex]\( f(g(x)) = x \)[/tex]
Let's go through each pair of functions to verify this.
### Pair A: [tex]\( f(x) = 3x - 4 \)[/tex] and [tex]\( g(x) = -3x + 4 \)[/tex]
Step 1: Check [tex]\( g(f(x)) \)[/tex]
[tex]\[ g(f(x)) = g(3x - 4) = -3(3x - 4) + 4 = -9x + 12 + 4 = -9x + 16 \][/tex]
Clearly, [tex]\( -9x + 16 \neq x \)[/tex].
Step 2: Check [tex]\( f(g(x)) \)[/tex]
[tex]\[ f(g(x)) = f(-3x + 4) = 3(-3x + 4) - 4 = -9x + 12 - 4 = -9x + 8 \][/tex]
Again, [tex]\( -9x + 8 \neq x \)[/tex].
Since both conditions fail, Pair A is not made up of inverse functions.
### Pair B: [tex]\( f(x) = \frac{x-3}{2} \)[/tex] and [tex]\( g(x) = 2x - 3 \)[/tex]
Step 1: Check [tex]\( g(f(x)) \)[/tex]
[tex]\[ g(f(x)) = g\left(\frac{x-3}{2}\right) = 2\left(\frac{x-3}{2}\right) - 3 = (x - 3) - 3 = x - 6 \][/tex]
Clearly, [tex]\( x - 6 \neq x \)[/tex].
Step 2: Check [tex]\( f(g(x)) \)[/tex]
[tex]\[ f(g(x)) = f(2x - 3) = \frac{2x - 3 - 3}{2} = \frac{2x - 6}{2} = x - 3 \][/tex]
Again, [tex]\( x - 3 \neq x \)[/tex].
Since both conditions fail, Pair B is not made up of inverse functions.
### Pair C: [tex]\( f(x) = 3x + 2 \)[/tex] and [tex]\( g(x) = 3x - 2 \)[/tex]
Step 1: Check [tex]\( g(f(x)) \)[/tex]
[tex]\[ g(f(x)) = g(3x + 2) = 3(3x + 2) - 2 = 9x + 6 - 2 = 9x + 4 \][/tex]
Clearly, [tex]\( 9x + 4 \neq x \)[/tex].
Step 2: Check [tex]\( f(g(x)) \)[/tex]
[tex]\[ f(g(x)) = f(3x - 2) = 3(3x - 2) + 2 = 9x - 6 + 2 = 9x - 4 \][/tex]
Again, [tex]\( 9x - 4 \neq x \)[/tex].
Since both conditions fail, Pair C is not made up of inverse functions.
### Pair D: [tex]\( f(x) = \frac{x - 4}{5} \)[/tex] and [tex]\( g(x) = 5x + 4 \)[/tex]
Step 1: Check [tex]\( g(f(x)) \)[/tex]
[tex]\[ g(f(x)) = g\left(\frac{x - 4}{5}\right) = 5\left(\frac{x - 4}{5}\right) + 4 = (x - 4) + 4 = x \][/tex]
Here, [tex]\( g(f(x)) = x \)[/tex].
Step 2: Check [tex]\( f(g(x)) \)[/tex]
[tex]\[ f(g(x)) = f(5x + 4) = \frac{5x + 4 - 4}{5} = \frac{5x}{5} = x \][/tex]
And here, [tex]\( f(g(x)) = x \)[/tex].
Since both conditions are satisfied, Pair D is made up of inverse functions.
Therefore, the pair of functions that are inverse functions is:
[tex]\[ \boxed{D} \][/tex]
