Question

Consider this function.

Answer 1

Answer:

3. Since the range of the original function is limited to y> 6, the domain of the inverse function is x ≥ 6.

Step-by-step explanation:

The domain of a function is the range of its inverse, and vice versa. The only answer choice that expresses this relationship is choice 3.

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Comment on the answer choice:

The slope of the function is undefined at x=4, so restricting the function domain to the portion with positive slope means the domain restriction of the function is x > 4. That also means the range restriction of the function is y > 6. The domain restriction of the inverse function is the same: x > 6, not x ≥ 6. The answer choice has an error.

Answer 2

Answer:

3. Since the range of the original function is limited to y> 6, the domain of the inverse function is x ≥ 6.

Step-by-step explanation:

Given absolute function,.

f(x) = |x-4|+6,

Since, an absolute function is defined for all real numbers,

So, the domain of f(x) is the set of all real numbers,

∴ Options (1) and (2) can not be true.

Now, for any real number,

|x-4| ≥ 0,

That is, f(x) ≥ 6,

Hence, the range ( possible value of output of a function ) of f(x) would be all real numbers greater than equal to 6,

I.e. Range of f(x) is, f(x) ≥ 6,

⇒ if y = f(x) ⇒ the range of the original function is limited to y > 6

∴ Option (4) can not be true,

Also, range of $\displaystyle f$ = Domain of $\displaystyle {f}^{-1}$

Hence, domain of $\displaystyle {f}^{-1}$ is x ≥ 6.