Question

Consider the function f(x)=2x sqrt(+5). If -1(x) is the inverse function of f(x), find

f^−1 (7).

a) f −1 (7)=2
b) f −1 (7)=1
c) f −1 (7)=3
d) f −1 (7)=4

Answer 1

To find the inverse of the function f(x)=2x * sqrt(x+5), replace f(x) with y and solve the equation for y. However, the resulting equation is a cubic equation and cannot be easily solved for y. Therefore, we cannot determine the value of f^(-1)(7).

Step-by-step explanation:

To find the inverse of a function, we can start by replacing f(x) with y in the original function. So, we have y = 2x * sqrt(x+5). To find the inverse, we need to swap the variables x and y and solve for y. So, x = 2y * sqrt(y+5). Now, we can solve this equation for y to find the inverse function.

First, divide both sides of the equation by 2: x/2 = y * sqrt(y+5). Next, square both sides of the equation to get rid of the square root: (x/2)^2 = y^2 * (y+5). Simplify each side of the equation: x^2/4 = y^2 * (y+5). Then, divide both sides of the equation by (y+5): x^2/4 / (y+5) = y^2. Multiply both sides of the equation by (y+5) to eliminate the fraction: (x^2/4) * (y+5) = y^2 * (y+5). Finally, distribute on the left side of the equation: (x^2/4) * y + (x^2/4) * 5 = y^3 + 5y^2.

This equation is a cubic equation, and solving for y can be quite complicated. So, it is not possible to find an explicit expression for the inverse function f^(-1)(x) in terms of x. Therefore, we cannot determine the value of f^(-1)(7).