Question

What is the inverse of f(x)=(2x+1) √2

​ for x≥−1/2​?
A) g(x)=(x²−1) /2
B) g(x)=(x²−2) /2
C) g(x)=(x²+1 )/2
D) g(x)=9x²+2) /2

Answer 1

The inverse of the function f(x)=(2x+1) \u221A2 for x \u2265 -1/2 is g(x)=(x2 - 1) / 2, following a step-by-step algebraic process.

Step-by-step explanation:

The inverse function of f(x) = (2x+1) \u221A2 for x \u2265 -1/2 is found by swapping the roles of x and y in the equation, then solving for y. Start by letting y = (2x+1) \u221A2 and rewrite it as x = (2y+1) \u221A2. Solve the equation step-by-step:

Divide both sides by \u221A2 to get x/\u221A2 = 2y+1.

Subtract 1 from both sides to get (x/\u221A2) - 1 = 2y.

Finally, divide by 2 to solve for y, resulting in y = ((x/\u221A2) - 1)/2.

Rewriting this with x replacing y gives us g(x) = ((x/\u221A2) - 1)/2. To simplify further, square \u221A2 to get 2 and multiply through the equation to eliminate the fraction under x, resulting in g(x) = (x2 - 1) / 2, which is answer choice A.