Consider the function and its inverse. and When comparing the functions using the equations, which conclusion can be made? The domain of f (x) is restricted to x ≥ 0, and the domain of f-1 (x) is restricted to x ≥ 0. The domain of f (x) is restricted to x ≥ 0, and the domain of f-1 (x) is restricted to x ≤ 0. The domain of f (x) is restricted to x ≤ 0, and the domain of f-1 (x) is restricted to x ≥ 4. The domain of f (x) is restricted to x ≤ 0, and the domain of f-1 (x) is restricted to x ≤ 4.

Step-by-step explanation:

The original function / displaystyle f/left (x/right) = {/left (x - 4/right) }^{2}f (x) = (x-4)

2

is not one-to-one, but the function is restricted to a domain of / displaystyle x/ge 4x≥4 or / displaystyle x/le 4x≤4 on which it is one-to-one.

Two graphs of f (x) = (x-4) ^2 where the first is when x>=4 and the second is when x<=4.

Figure 5

To find the inverse, start by replacing / displaystyle f/left (x/right) f (x) with the simple variable y.

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

y

=

(

x

-

4

)

2

Interchange

x

and

y

.

x

=

(

y

-

4

)

2

Take the square root

.

±

√

x

=

y

-

4

Add

4

to both sides

.

4

±

√

x

=

y

This is not a function as written. We need to examine the restrictions on the domain of the original function to determine the inverse. Since we reversed the roles of x and y for the original f (x), we looked at the domain: the values x could assume. When we reversed the roles of x and y, this gave us the values y could assume. For this function, / displaystyle x/ge 4x≥4, so for the inverse, we should have / displaystyle y/ge 4y≥4, which is what our inverse function gives.

The domain of the original function was restricted to / displaystyle x/ge 4x≥4, so the outputs of the inverse need to be the same, / displaystyle f/left (x/right) / ge 4f (x) ≥4, and we must use the + case:

/displaystyle {f}^{-1}/left (x/right) = 4+/sqrt{x}f

-1

(x) = 4+√

x





The domain of the original function was restricted to / displaystyle x/le 4x≤4, so the outputs of the inverse need to be the same, / displaystyle f/left (x/right) / le 4f (x) ≤4, and we must use the - case:

/displaystyle {f}^{-1}/left (x/right) = 4-/sqrt{x}f

-1

(x) = 4-√

x