To determine which of the given statements are true about both [tex]\( f(x) = \frac{1}{x} \)[/tex] and [tex]\( f(x) = \log(x) \)[/tex], let us examine each statement in detail:
Statement I: They are undefined at [tex]\( x = 0 \)[/tex]
- For [tex]\( f(x) = \frac{1}{x} \)[/tex], the function is undefined at [tex]\( x = 0 \)[/tex] because division by zero is undefined.
- For [tex]\( f(x) = \log(x) \)[/tex], the function is also undefined at [tex]\( x = 0 \)[/tex] because the logarithm of zero is undefined.
Therefore, this statement is true for both functions.
Statement II: Their domain is only positive numbers
- For [tex]\( f(x) = \frac{1}{x} \)[/tex], the domain includes all real numbers except [tex]\( x = 0 \)[/tex]. This means [tex]\( x \)[/tex] can be both positive and negative, thus the domain is [tex]\( x \in \mathbb{R}\backslash\{0\} \)[/tex].
- For [tex]\( f(x) = \log(x) \)[/tex], the domain is only positive numbers because the logarithm function is only defined for [tex]\( x > 0 \)[/tex].
Since [tex]\( f(x) = \frac{1}{x} \)[/tex] includes negative numbers in its domain, this statement is false.
Statement III: They both have two curves
- The graph of [tex]\( f(x) = \frac{1}{x} \)[/tex] has two branches, one in the first quadrant (where [tex]\( x > 0 \)[/tex]) and one in the third quadrant (where [tex]\( x < 0 \)[/tex]).
- The graph of [tex]\( f(x) = \log(x) \)[/tex] has only one branch, which is in the first quadrant (where [tex]\( x > 0 \)[/tex]).
Since [tex]\( f(x) = \log(x) \)[/tex] does not have two curves, this statement is false.
Based on the evaluations:
- Statement I is true.
- Statement II is false.
- Statement III is false.
Thus, the correct answer is:
[tex]\[ a. \text{I only} \][/tex]
