Question

Two functions are represented below. Which function has a domain that contains the domain of the other function as a

Answer 1

Answer:

$\\ h(x) = x-2\;contains\;the\;domain\;of\;function\;f(x) = -log(x-2)-3$

Step-by-step explanation:

The domain of a function contains all possible values that it can take.

We can analyze the function:

$\\ f(x) = -log(x-2)-3$ [1]

And we can see that there are three functions in [1]:

$h(x) = x-2$ and $g(x) = -log(x) - 3$, and

$\\ f(x) = g(h(x))$ or $\\ f(x) = -log(x-2)-3$.

This operation is called function composition.

For every x, function h(x) can take any value, so the domain (all possible values this function can take) is $\\ -\infty\;to\;\infty$ (all real numbers).

However, function g(x) is restricted to positive values only, since log function is not defined for negative numbers and 0.

Therefore, domain for f(x) are restricted to those ones that comply with the above restriction (x > 0). So, for

$\\ f(x) = -log(x-2) - 3$

Possible values are those which:

$\\ x - 2 > 0$ or $\\ x > 2$

In words, the domain of f(x) are all values greater than 2 (but not equal to 2) or all positive real values greater than 2.

Then, the function f(x) has a domain that is a subset of function h(x), that is, in function h(x) = x - 2, x can takes any possible value from -(infinity) to infinity, whereas function f(x) can only takes those values greater than two (but not equal 2) to infinity, which makes the domain of f(x)  a subset of the possible values of function h(x).