Question

What is the inverse

Answer 1

Answer:

Third choice

$\frac{\sqrt{2x}}{2} + 5, x\ge0$

Step-by-step explanation:

Definition

An inverse function is defined as a function, which can reverse into another function.

Thus, if we have a function f(x) which evaluates to k at x = a or in other words f(a) = k, then the inverse of that function indicated by f⁻¹(x) will be such that f⁻¹(k) = a. This is the same as stating that f⁻¹(f(a)) = a

In order the find the in verse of a function f(x) the following are the steps

• Set y = f(x)
• Switch x and y
• Solve for y

Given  $f(x) = 2 \left(x - 5\right)^{2}$, set it equal to y

$y=2 \left(x - 5\right)^{2}$  

Swap the variables x and y

$x=2 \left(y - 5\right)^{2}$  ==> $(x - 5)^2 = \frac{x}{2}$

$x - 5 = \pm \sqrt\frac{x}{2}}\\\\x = \pm \sqrt\frac{x}{2}} + 5$

Solve for y. There are two possible solutions

$y= \frac{\sqrt{2} \sqrt{x}}{2} + 5 = \frac{\sqrt{2x}}{2} + 5$

$y=- \frac{\sqrt{2} \sqrt{x}}{2} + 5$ = $- \frac{\sqrt{2x}}{2} + 5$

The only matching choice are choices 1 and 3. In order to determine if it is the first or the third, plug in 0 and see if the inverse value results in a real number

Substituting 0 for x in $\frac{\sqrt{2x}}{2} + 5$ gives us 0 + 5 = 5 which is a valid real

So the domain of the function are all values ≥ 0

Hence choice 3, not choice 1