If `f :R ->R` , `f(x)=x^3 +3`,and `g:R->R`,`g(x)=2x + 1`, then `f^(-1)(g^(-1)(23))` equals

Options

A. 2
B. 3
C. `(14)^(1//3)`
D. `(15)^(1//3)`

Explanation

To solve the problem, we need to find \( f^{-1}(g^{-1}(23)) \) given the functions \( f(x) = x^3 + 3 \) and \( g(x) = 2x + 1 \).

### Step 1: Find \( g^{-1}(23) \)

The function \( g(x) = 2x + 1 \). To find the inverse, we set \( y = g(x) \):
\[ y = 2x + 1 \]
Now, solve for \( x \):
\[ y - 1 = 2x \\ x = \frac{y - 1}{2} \]
Thus, the inverse function is:
\[ g^{-1}(y) = \frac{y - 1}{2} \]
Now, substitute \( 23 \) into \( g^{-1} \):
\[ g^{-1}(23) = \frac{23 - 1}{2} = \frac{22}{2} = 11 \]

### Step 2: Find \( f^{-1}(11) \)

Next, we need to find \( f^{-1}(x) \) for \( f(x) = x^3 + 3 \). Set \( y = f(x) \):
\[ y = x^3 + 3 \]
Now, solve for \( x \):
\[ y - 3 = x^3 \\ x = (y - 3)^{1/3} \]
Thus, the inverse function is:
\[ f^{-1}(y) = (y - 3)^{1/3} \]
Now, substitute \( 11 \) into \( f^{-1} \):
\[ f^{-1}(11) = (11 - 3)^{1/3} = 8^{1/3} = 2 \]

### Final Answer

Thus, the value of \( f^{-1}(g^{-1}(23)) \) is:
\[ \boxed{2} \]