If `f(x)` is an invertible function and `g(x)=2f(x)+5,` then the value of `g^(-1)(x)i s` (a) `2f^(-1)(x)-5` (b) `1/(2f^(-1)(x)+5)` `1/2f^(-1)(x)+5` (d) `f^(-1)((x-5)/2)`

Options

A. `2f^(-1)(x)-5`
B. `(1)/(2f^(-1)(x)+5)`
C. `(1)/(2) f^(-1)(x)+5`
D. `f^(-1)((x-5)/(2))`

Explanation

To find the value of \( g^{-1}(x) \) given that \( g(x) = 2f(x) + 5 \), we can follow these steps:

### Step 1: Write the equation for \( g(g^{-1}(x)) \)
We know that for any function \( g \) and its inverse \( g^{-1} \), the following holds true:
\[ g(g^{-1}(x)) = x \]
Substituting \( g(x) \):
\[ g(g^{-1}(x)) = 2f(g^{-1}(x)) + 5 \]
Thus, we have:
\[ 2f(g^{-1}(x)) + 5 = x \]

### Step 2: Isolate \( f(g^{-1}(x)) \)
To isolate \( f(g^{-1}(x)) \), we subtract 5 from both sides:
\[ 2f(g^{-1}(x)) = x - 5 \]

### Step 3: Divide by 2
Now, we divide both sides by 2:
\[ f(g^{-1}(x)) = \frac{x - 5}{2} \]

### Step 4: Apply the inverse of \( f \)
Since \( f \) is an invertible function, we can apply \( f^{-1} \) to both sides:
\[ g^{-1}(x) = f^{-1}\left(\frac{x - 5}{2}\right) \]

### Conclusion
Thus, the value of \( g^{-1}(x) \) is:
\[ g^{-1}(x) = f^{-1}\left(\frac{x - 5}{2}\right) \]

### Final Answer
The correct option is:
(d) \( f^{-1}\left(\frac{x - 5}{2}\right) \)

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