If f: R `rarr` -R. g : R `rarr` R be two functions given by f(x) = 2x-3 and g(x)=`x^(3) + 5`. then `(fog)^(-1)` (x) is equal to:

Options

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

Answer

To find \((f \circ g)^{-1}(x)\), we need to first determine the composition of the functions \(f\) and \(g\), and then find the inverse of that composition.

### Step 1: Define the functions
We have:
- \(f(x) = 2x - 3\)
- \(g(x) = x^3 + 5\)

### Step 2: Find the composition \(f(g(x))\)
To find \(f(g(x))\), we substitute \(g(x)\) into \(f(x)\):
\[ f(g(x)) = f(x^3 + 5) \]
Now, replace \(x\) in \(f(x)\) with \(x^3 + 5\):
\[ f(g(x)) = 2(x^3 + 5) - 3 \]
Distributing the 2:
\[ = 2x^3 + 10 - 3 \]
Simplifying:
\[ = 2x^3 + 7 \]

### Step 3: Set \(y = f(g(x))\) and solve for \(x\)
Let \(y = 2x^3 + 7\). We need to express \(x\) in terms of \(y\):
\[ y - 7 = 2x^3 \]
Dividing both sides by 2:
\[ x^3 = \frac{y - 7}{2} \]
Taking the cube root of both sides:
\[ x = \sqrt[3]{\frac{y - 7}{2}} \]

### Step 4: Write the inverse function
Thus, we have:
\[ (f \circ g)^{-1}(y) = \sqrt[3]{\frac{y - 7}{2}} \]
To express it in terms of \(x\):
\[ (f \circ g)^{-1}(x) = \sqrt[3]{\frac{x - 7}{2}} \]

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

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