Maths · Inverse functions
If =x^{2}-x+5, x>\frac{1}{2}, \) and \) is its inverse function, then \) equals:
If \( f(x)=x^{2}-x+5, x>\frac{1}{2}, \) and \( g(x) \) is its inverse function, then \( g^{\prime}(7) \) equals:
- A. \( -\frac{1}{3} \)
- B. \( \frac{1}{13} \)
- C. \( \frac{1}{3} \)
- D. \( -\frac{1}{13} \)
Step-by-step solution
Since g is the inverse of f, we have g'(7) = 1 / f'(g(7)). Solve f(x)=7: x^2 - x + 5 = 7 => x^2 - x - 2 = 0 => (x-2)(x+1)=0 => x=2 (since x>1/2). So g(7)=2. Then f'(x)=2x-1, so f'(2)=3. Thus g'(7)=1/3.