如何构造满足截距与渐近线条件的有理函数$h(x)$？
Let’s break this down step by step—each requirement gives us clear clues about how to build the rational function $h(x)$:
1. Account for x-intercepts at $x=1$ and $x=3$
X-intercepts occur where the function equals 0, which means the numerator must be zero at these points. We add linear factors to the numerator that vanish when $x=1$ and $x=3$:
- For $x=1$: Factor is $(x - 1)$
- For $x=3$: Factor is $(x - 3)$
Our starting numerator is: $(x - 1)(x - 3)$
2. Add vertical asymptotes at $x=2$ and $x=5$
Vertical asymptotes happen where the function is undefined (denominator equals 0) and the numerator doesn’t equal 0 at those points. We add linear factors to the denominator that vanish at $x=2$ and $x=5$:
- For $x=2$: Factor is $(x - 2)$
- For $x=5$: Factor is $(x - 5)$
Our starting denominator is: $(x - 2)(x - 5)$
3. Set the horizontal asymptote to $y=4$
A horizontal asymptote of $y=4$ means as $x \to \pm\infty$, the function approaches 4. For rational functions, this occurs when the numerator and denominator have the same degree, and the ratio of their leading coefficients is 4.
Right now, both numerator and denominator are degree 2 (expanding them gives highest power $x^2$). The leading coefficient of the numerator is 1, and the denominator’s leading coefficient is 1. To get a ratio of 4, we multiply the entire numerator by 4.
Final Function
Combining all these pieces, the simplest function that meets all your requirements is:
$$h(x) = \frac{4(x - 1)(x - 3)}{(x - 2)(x - 5)}$$
Quick Verification
- X-intercepts: Plugging $x=1$ or $x=3$ makes the numerator 0, so $h(x)=0$ ✔️
- Vertical asymptotes: Plugging $x=2$ or $x=5$ makes the denominator 0, and the numerator isn’t zero at these points (e.g., $4(2-1)(2-3) = -4 \neq 0$) ✔️
- Horizontal asymptote: As $x$ grows large, lower-degree terms become irrelevant, so we’re left with $\frac{4x2}{x2} = 4$ ✔️
Note: You could multiply the numerator or denominator by any non-zero constant (like 2 or $\frac{1}{3}$) without changing intercepts or asymptotes—this just scales the function. The form above is the most straightforward version.
内容的提问来源于stack exchange，提问作者CodeMan5000
