If `fQ to Q f(x)=2x,g,Q to Q, g (x)=x+2` then value of `(fog)^(-1)(20)` is

Options

A. 5
B. `-8`
C. 4
D. 8

Answer

To solve the problem, we need to find the value of \((f \circ g)^{-1}(20)\). Let's break this down step by step.

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

### Step 2: Find \( f \circ g \)
The composition of the functions \( f \) and \( g \) is defined as:
\[ (f \circ g)(x) = f(g(x)) \]
Substituting \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(x + 2) = 2(x + 2) = 2x + 4 \]
Thus, we have:
\[ f \circ g = 2x + 4 \]

### Step 3: Set up the equation for the inverse
We need to find \((f \circ g)^{-1}(20)\). To do this, we first set \( y = f \circ g(x) \):
\[ y = 2x + 4 \]
Now, we need to solve for \( x \) in terms of \( y \):
\[ y = 2x + 4 \]
Subtract 4 from both sides:
\[ y - 4 = 2x \]
Now, divide by 2:
\[ x = \frac{y - 4}{2} \]

### Step 4: Find the inverse function
Thus, the inverse function is:
\[ (f \circ g)^{-1}(y) = \frac{y - 4}{2} \]

### Step 5: Calculate \((f \circ g)^{-1}(20)\)
Now, we substitute \( y = 20 \) into the inverse function:
\[ (f \circ g)^{-1}(20) = \frac{20 - 4}{2} = \frac{16}{2} = 8 \]

### Final Answer
The value of \((f \circ g)^{-1}(20)\) is \( \boxed{8} \).

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