If the functions of `f` and `g` are defined by `f(x)=3x-4` and `g(x)=2+3x` then `g^(-1)(f^(-1)(5))`

Options

A. 1
B. `1//2`
C. `1//3`
D. `1//4`

Answer

To solve the problem, we need to find \( g^{-1}(f^{-1}(5)) \) where the functions are defined as \( f(x) = 3x - 4 \) and \( g(x) = 2 + 3x \).

### Step 1: Find \( f^{-1}(x) \)

We start by setting \( y = f(x) \):
\[ y = 3x - 4 \]
Now, we solve for \( x \):
\[ 3x = y + 4 \]
\[ x = \frac{y + 4}{3} \]
Thus, we can express the inverse function as:
\[ f^{-1}(x) = \frac{x + 4}{3} \]

### Step 2: Find \( f^{-1}(5) \)

Now, we need to evaluate \( f^{-1}(5) \):
\[ f^{-1}(5) = \frac{5 + 4}{3} = \frac{9}{3} = 3 \]

### Step 3: Find \( g^{-1}(x) \)

Next, we find \( g^{-1}(x) \). We start by setting \( y = g(x) \):
\[ y = 2 + 3x \]
Now, we solve for \( x \):
\[ 3x = y - 2 \]
\[ x = \frac{y - 2}{3} \]
Thus, we can express the inverse function as:
\[ g^{-1}(x) = \frac{x - 2}{3} \]

### Step 4: Find \( g^{-1}(f^{-1}(5)) \)

Now we need to evaluate \( g^{-1}(f^{-1}(5)) \):
\[ g^{-1}(3) = \frac{3 - 2}{3} = \frac{1}{3} \]

### Final Answer

Thus, the final answer is:
\[ g^{-1}(f^{-1}(5)) = \frac{1}{3} \]

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