Answer:
Step-by-step explanation:
The domain of a function contains all possible values that it can take.
We can analyze the function:
[1]
And we can see that there are three functions in [1]:
and
, and
or
.
This operation is called <em>function composition</em>.<em> </em>
For every <em>x</em>, function <em>h(x)</em> can take any value, so the domain (all possible values this function can take) is (all real numbers).
However, function <em>g(x)</em> is restricted to <em>positive values</em> only, since <em>log</em> function is <em>not</em> defined for <em>negative numbers</em> and 0.
Therefore, domain for <em>f(x)</em> are <em>restricted</em> to those ones that comply with the above restriction (x > 0). So, for
Possible values are those which:
or
In words, the domain of <em>f(x)</em> are all values greater than 2 (but not equal to 2) or all positive real values greater than 2.
Then, the function <em>f(x)</em> has a domain that is a <em>subset</em> of function <em>h(x)</em>, that is, in function <em>h(x) = </em>x - 2, <em>x</em> can takes any possible value from -(<em>infinity</em>) to <em>infinity</em>, whereas function <em>f(x)</em> can <em>only</em> takes those values greater than two (but not equal 2) to <em>infinity</em>, which makes the domain of <em>f(x)</em> a subset of the possible values of function <em>h(x)</em>.