Let `f:R rarr R`, `g : R rarr R` be two function given by `f(x) =5x-4` and `g(x)` =`x^(3)+7` then `(fog)^(-1) (x)` equals

Options

A. `((x+31)/(5))^(1//3)`
B. `((x-31)/(5))^(1//3)`
C. `(x-5)/(7)^(1//3)`
D. `(x-31/5)^(1//3)`

Answer

To find \((f \circ g)^{-1}(x)\), we will follow these steps:

### Step 1: Define the functions
We have two functions:
- \( f(x) = 5x - 4 \)
- \( g(x) = x^3 + 7 \)

### Step 2: Find \( f(g(x)) \)
We need to compute \( f(g(x)) \):
\[ f(g(x)) = f(x^3 + 7) \]
Substituting \( g(x) \) into \( f(x) \):
\[ f(g(x)) = 5(x^3 + 7) - 4 \]
Now simplify:
\[ = 5x^3 + 35 - 4 \]
\[ = 5x^3 + 31 \]

### Step 3: Set \( y = f(g(x)) \)
Let \( y = f(g(x)) \):
\[ y = 5x^3 + 31 \]

### Step 4: Solve for \( x \) in terms of \( y \)
Rearranging the equation to solve for \( x \):
\[ y - 31 = 5x^3 \]
\[ x^3 = \frac{y - 31}{5} \]
Taking the cube root on both sides:
\[ x = \sqrt[3]{\frac{y - 31}{5}} \]

### Step 5: Write the inverse function
Thus, the inverse function \((f \circ g)^{-1}(x)\) is:
\[ (f \circ g)^{-1}(x) = \sqrt[3]{\frac{x - 31}{5}} \]

### Final Answer
\[ (f \circ g)^{-1}(x) = \sqrt[3]{\frac{x - 31}{5}} \]

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