If f is a function defined as `f ( x) = x^(2)-x+5, f: ((1)/(2) , oo) rarr ((19)/(4) , oo)`, and g(x) is its inverse function, then g'(7) is equal to

Options

A. `- (1)/(13)`
B. `(1)/(13)`
C. `(1)/(3)`
D. `-(1)/( 3)`

Answer

To find \( g'(7) \) where \( g(x) \) is the inverse of the function \( f(x) = x^2 - x + 5 \), we will follow these steps:

### Step 1: Identify the function and its inverse
We are given:
\[ f(x) = x^2 - x + 5 \]
and we need to find \( g'(7) \), where \( g(x) \) is the inverse of \( f(x) \).

### Step 2: Find the inverse function
To find the inverse function \( g(x) \), we set \( y = f(x) \):
\[ y = x^2 - x + 5 \]
Rearranging gives us:
\[ x^2 - x + (5 - y) = 0 \]
This is a quadratic equation in \( x \). The roots can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -1, c = 5 - y \).

### Step 3: Substitute values into the quadratic formula
Substituting into the formula:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (5 - y)}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{1 \pm \sqrt{1 - 20 + 4y}}{2} = \frac{1 \pm \sqrt{4y - 19}}{2} \]

### Step 4: Determine the correct root
Since \( f(x) \) is increasing on the interval \( \left(\frac{1}{2}, \infty\right) \), we take the positive root:
\[ g(y) = \frac{1 + \sqrt{4y - 19}}{2} \]

### Step 5: Differentiate the inverse function
To find \( g'(x) \), we differentiate \( g(y) \):
\[ g'(y) = \frac{1}{2} \cdot \frac{1}{2\sqrt{4y - 19}} \cdot 4 = \frac{2}{\sqrt{4y - 19}} \]

### Step 6: Evaluate \( g'(7) \)
Now we substitute \( y = 7 \) into \( g'(y) \):
\[ g'(7) = \frac{2}{\sqrt{4 \cdot 7 - 19}} = \frac{2}{\sqrt{28 - 19}} = \frac{2}{\sqrt{9}} = \frac{2}{3} \]

### Final Answer
Thus, the value of \( g'(7) \) is:
\[ \boxed{\frac{2}{3}} \]