Question

What is the inverse of the function f (x) = x2 – 49?

O A -'(x) = *V* + 7 for x > 0
B. 8-1(x) = +Vx+ 49 for x > 0
Oc5-1(x) = +Vx+7 for x > -7
D. f-1(x) = -VX – 49 for x > 49
O
E. F-1(x) = +Vx+49 for x>-49

Answer 1

The inverse of the function f(x) = x^2 − 49 is g(x) = +√x+49 for x ≥ −49, which corresponds to Option E.

Step-by-step explanation:

The inverse of a function f(x) essentially reverses the effect of f(x). In the case of f(x) = x^2−49, finding the inverse function involves exchanging the roles of x and y and then solving for y. This process typically involves finding the square root of x. Since the original function involves a square, which is an even function, we need to take into account the sign of the original x when considering its inverse.

To find the inverse, we set y = x^2 − 49 and then solve for x:

1. x = y + 49
2. x = ±√y + 49

However, because we are looking for a function, we must choose only one of the two possible square roots for each input. Since the original function is only increasing, its inverse will only take the non-negative root for x values greater than or equal to 49. Thus, the inverse function is g(x) = +√x+49 for x ≥ 49.

In context of the provided options, the correct answer would be Option E: f−1(x) = +√x+49 for x≥−49.