📚 G - The Reciprocal of a Function | 函数的倒数变换
ral to the IB Mathematics curriculum, especially in studying rational functions, asymptotes, and graph sketching. Understanding reciprocals allows us to explore how the output values of f(x) are inverted, leading to dramatic changes in shape, domain, and range. In this article, we examine the definition, graphical properties, and common applications of reciprocal functions.
函数的倒数是一种变换，生成新函数 g(x) = 1/f(x)。这一概念在 IB 数学课程中至关重要，尤其是在研究有理函数、渐近线以及图形绘制时。理解倒数能让我们探索 f(x) 的输出值如何被“翻转”，从而导致图像形状、定义域和值域的剧烈变化。本文将研究倒数函数的定义、图形性质及其常见应用。
1. Definition and Notation | 定义与符号
(x), where the denominator is not zero. It is often denoted as (1/f)(x). The reciprocal is not (x) = 1/x. We must always state the domain: r(x) is defined for all x such that f(x) <> 0.
函数的倒数是由 r(x) = 1/f(x) 定义的函数，其中分母不为零。它通常记作 (1/f)(x)。倒数并不是反函数，而是一种完全不同的变换。例如，若 f(x) = x，则其倒数为 r(x) = 1/x。我们总是需要说明定义域：r(x) 在所有使得 f(x) <> 0 的 x 上有定义。
2. Domain and Vertical Asymptotes | 定义域与垂直渐近线
hange sign (e.g., a double root), the reciprocal tends to +∞ (or -∞) on both sides. For instance, with f(x) = x², zero is a double root, and 1/x² → +∞ as x → 0 from either side.
在构造 1/f(x) 时，任何使 f(x) = 0 的 x 都必须从定义域中去除。这些被排除的值就成为倒数函数图像的垂直渐近线。若 f 连续且在该零点处变号，倒数会在该点一侧趋于 +∞，另一侧趋于 -∞。若 f 不变号（例如二重零点），倒数在两侧都趋于 +∞（或都趋于 -∞）。例如，取 f(x) = x²;，零点是二重根，当 x 从任意一侧趋于 0 时，1/x² → +∞。
3. Horizontal Asymptotes and End Behaviour | 水平渐近线与末端行为
The behaviour of 1/f(x) as x → ±∞ depends on the limit of f(x). If |f(x)| . If f(x) oscillates, the reciprocal may have no horizontal asymptote. Otherwise, consider end behaviour: as the magnitude of f(x) becomes very large, the reciprocal becomes very small, hugging the x-axis.
当 x → ±∞ 时，1/f(x) 的行为取决于 f(x) 的极限。如果 |f(x)| → ∞，那么 1/f(x) → 0，因此直线 y = 0 为水平渐近线。若 f(x) 趋近于一个有限非零值 L，则倒数趋于 1/L，得到水平渐近线 y = 1/L。如果 f(x) 振荡，倒数可能没有水平渐近线。通常，末端行为是：当 f(x) 的绝对值变得很大时，倒数非常接近 0，紧贴 x 轴。
4. Sketching Reciprocal Graphs from f(x) | 从 f(x) 画出倒数函数图像
To sketch y = 1/f(x) given the graph of y = f(x), follow these steps:
- Identify zeros of f: draw vertical dashed lines at these x-values (asymptotes).
- Mark invariant points where f(x) = 1 or f(x) = -1; these points lie on both graphs.
- Where f(x) > 0, the reciprocal is also positive. Where f(x) < 0, the reciprocal is negative.
- In intervals where f(x) is large in magnitude, the reciprocal is close to 0; where f(x) is close to 0, the reciprocal is large in magnitude.
- Near vertical asymptotes, decide the direction (±∞) based on the sign of f(x) on each side.
- Consider monotonicity: if f is increasing and stays positive, then 1/f is decreasing; but be careful with sign changes.
要依据 y = f(x) 的图像绘制 y = 1/f(x)，可以遵循以下步骤：
- 确定 f 的零点：在这些 x 值处画出垂直虚线（渐近线）。
- 标记 f(x) = 1 或 f(x) = -1 的不动点；这些点同时位于两个图像上。
- 在 f(x) > 0 的区间，倒数也为正；在 f(x) < 0 的区间，倒数为负。
- 在 f(x) 绝对值很大的区间，倒数接近 0；在 f(x) 接近 0 的区间，倒数绝对值很大。
- 在垂直渐近线附近，根据两侧 f(x) 的符号确定倒数的趋向（±∞）。
- 考虑单调性：如果 f 递增且恒为正，则 1/f 递减；但要注意符号变化。
5. Key Features: Invariant Points and Sign | 关键特征：不变点与符号
, these points remain unchanged under the reciprocal transformation. They are called invariant points. The sign of 1/f(x) is identical to the sign of f(x) because the reciprocal preserves sign. This property helps determine in which regions the reciprocal graph lies above or below the x-axis. There are no x-intercepts for the reciprocal unless f(x) has a vertical asymptote, which is impossible since f is a function; thus the reciprocal never crosses the x-axis unless f itself tends to infinity, in which case it approaches 0 asymptotically.
f(x) = 1 或 f(x) = -1 的点非常特殊：因为 1/1 = 1 且 1/(-1) = -1，这些点在倒数变换下保持不变，称为不变点。1/f(x) 的符号与 f(x) 的符号完全相同，因为倒数不改变符号。这一性质有助于确定倒数图像在 x 轴上方还是下方的区域。倒数函数没有 x 轴截距，除非 f(x) 本身有垂直渐近线（那不可能，因为 f 是函数）；因此倒数图像永远不会穿越 x 轴，只会渐近地接近 0。
6. Reciprocal of Linear Functions | 线性函数的倒数
> -b/a, f(x) > 0 so the curve is in the first quadrant; for x < -b/a, it is in the third quadrant. If a < 0, the branches lie in the second and fourth quadrants. The axes of symmetry are lines with slope ±1 through the asymptote intersection.
考虑线性函数 f(x) = ax + b，其中 a ≠ 0。其倒数为
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图