Let `f:R to R, g: R to R` be two functions given by `f(x)=2x-3,g(x)=x^(3)+5`. Then `(fog)^(-1)` is equal to

Options

A. `((x+7)/(2))^(1//3)`
B. `(x-(7)/(2))^(1//3)`
C. `((x-2)/(7))^(1//3)`
D. `((x-7)/(2))^(1//3)`

Explanation

To solve the problem, we need to find the inverse of the composition of the functions \( f \) and \( g \), denoted as \( (f \circ g)^{-1} \).

### Step-by-Step Solution:

1. **Identify the Functions**:
- We have two functions defined as:
\[ f(x) = 2x - 3 \]
\[ g(x) = x^3 + 5 \]

2. **Find the Composition \( f(g(x)) \)**:
- We need to compute \( f(g(x)) \):
\[ f(g(x)) = f(x^3 + 5) \]
- Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = 2(x^3 + 5) - 3 \]
- Simplify the expression:
\[ = 2x^3 + 10 - 3 = 2x^3 + 7 \]

3. **Set \( y = f(g(x)) \)**:
- Let \( y = 2x^3 + 7 \).

4. **Solve for \( x \) in terms of \( y \)**:
- Rearranging the equation:
\[ y - 7 = 2x^3 \]
- Divide both sides by 2:
\[ x^3 = \frac{y - 7}{2} \]
- Take the cube root of both sides:
\[ x = \sqrt[3]{\frac{y - 7}{2}} \]

5. **Express \( (f \circ g)^{-1}(y) \)**:
- We have found that:
\[ (f \circ g)^{-1}(y) = \sqrt[3]{\frac{y - 7}{2}} \]

6. **Replace \( y \) with \( x \)**:
- To find \( (f \circ g)^{-1}(x) \):
\[ (f \circ g)^{-1}(x) = \sqrt[3]{\frac{x - 7}{2}} \]

### Final Answer:
\[ (f \circ g)^{-1}(x) = \sqrt[3]{\frac{x - 7}{2}} \]