Mathematics

Mathematics

Problem

For the function f(x)=−3x+1f(x)=\dfrac{-3}{x+1}, consider the following statements: P: lim⁡x→+∞f(x)=0\displaystyle \lim_{x\to +\infty}f(x)=0 Q: The range of ff is R∖{0}\mathbb{R}\setminus\{0\}. R: The point (−2,3)(-2,3) lies on the graph of ff. Which one of the following options is correct? a) P and Q are true\text{P and Q are true} b) All P, Q and R are true\text{All P, Q and R are true} c) P and R are true\text{P and R are true} d) Q and R are true\text{Q and R are true}
Intuitive explanation

This problem asks you to evaluate three claims about a rational function by checking limits, range, and points on the graph. The key insight is that rational functions like this one have predictable behavior: as xx grows large, the fraction shrinks toward zero; the denominator creates a "forbidden" output value that the function can never reach; and you can verify any point by substitution. Test each statement independently using these tools—no fancy algebra needed, just careful reasoning about what the function actually does.

Intuitive explanation

This problem asks you to evaluate three claims about a rational function by checking limits, range, and points on the graph. The key insight is that rational functions like this one have predictable behavior: as xx grows large, the fraction shrinks toward zero; the denominator creates a "forbidden" output value that the function can never reach; and you can verify any point by substitution. Test each statement independently using these tools—no fancy algebra needed, just careful reasoning about what the function actually does.

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