To determine which statement is logically equivalent to [tex]\( q \rightarrow p \)[/tex] (if the graphs of two functions intersect at exactly one point, then two linear functions have different coefficients of [tex]\( x \)[/tex]), we need to find the contrapositive, which is logically equivalent to the original implication.
The contrapositive of [tex]\( q \rightarrow p \)[/tex] is [tex]\( \neg p \rightarrow \neg q \)[/tex].
Let's break down the contrapositive step-by-step:
1. Identify [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
- [tex]\( p \)[/tex]: Two linear functions have different coefficients of [tex]\( x \)[/tex].
- [tex]\( q \)[/tex]: The graphs of two functions intersect at exactly one point.
2. State [tex]\( \neg p \)[/tex] and [tex]\( \neg q \)[/tex]:
- [tex]\( \neg p \)[/tex]: Two linear functions have the same coefficients of [tex]\( x \)[/tex].
- [tex]\( \neg q \)[/tex]: The graphs of two functions do not intersect at exactly one point.
3. Formulate the contrapositive [tex]\( \neg p \rightarrow \neg q \)[/tex]:
- If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of the two functions do not intersect at exactly one point.
Next, we need to match this with the given statements:
1. If two linear functions have different coefficients of [tex]\( x \)[/tex], then the graphs of the two functions intersect at exactly one point.
2. If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of the two linear functions do not intersect at exactly one point.
3. If the graphs of two functions do not intersect at exactly one point, then the two linear functions have the same coefficients of [tex]\( x \)[/tex].
4. If the graphs of two functions intersect at exactly one point, then the two linear functions have the same coefficients of [tex]\( x \)[/tex].
From the given statements, statement 2 matches the contrapositive [tex]\( \neg p \rightarrow \neg q \)[/tex]:
- If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of the two linear functions do not intersect at exactly one point.
Therefore, the statement that is logically equivalent to [tex]\( q \rightarrow p \)[/tex] is:
Statement 2: If two linear functions have the same coefficients of [tex]\( x \)[/tex], then the graphs of the two linear functions do not intersect at exactly one point.
