关于函数的倒数与反函数的概念辨析及相关性质验证问询

Hey Heidi, great question—this is such a common mix-up, so let’s break it down clearly using that reflection over $y=x$ you mentioned, plus concrete examples to make everything click.

First, let’s talk about inverse functions and the $y=x$ reflection #

• An inverse function is all about undoing what the original function does. If you have a function $f(x)$ that takes an input $x$ and outputs $y$, its inverse $f^{-1}(y)$ takes that $y$ and gives you back the original $x$.
• The reflection over the line $y=x$ is exactly how we visualize this relationship: every point $(a, b)$ on the graph of $f(x)$ gets flipped to $(b, a)$ on the graph of $f^{-1}(x)$. But a quick caveat: this only works if $f(x)$ is one-to-one (passes the horizontal line test). If it’s not, the reflection will be a relation, not a function. For example:
  • $f(x) = x^2$ isn’t one-to-one over all real numbers—its reflection over $y=x$ is $x = y^2$, a parabola opening to the right that fails the vertical line test (so it’s not a function). But if we restrict $f(x)$ to non-negative reals, its inverse becomes $f^{-1}(x) = \sqrt{x}$, which is a valid function and does match the reflection.

Now, the reciprocal of a function—this is a totally different thing #

• The reciprocal of $f(x)$ is just $\frac{1}{f(x)}$ (as long as $f(x) \neq 0$). This is a multiplicative inverse, meaning multiplying $f(x)$ by its reciprocal gives you 1 for all valid $x$.
• This has no connection to reflecting over $y=x$ at all. Let’s use a simple example to drive this home:
  • Take $f(x) = 2x$. Its reciprocal is $\frac{1}{2x}$, which graphs as a hyperbola. Its inverse, though, is $f^{-1}(x) = \frac{x}{2}$—a straight line that’s the exact reflection of $2x$ over $y=x$. You can see immediately these are nothing alike.

Quick recap to keep them straight #

• Inverse function: Undoes the input-output mapping of the original function; visualized by reflecting over $y=x$ (only works for one-to-one functions).
• Reciprocal: Multiplicative inverse—gives a function that when multiplied by the original equals 1; no link to the $y=x$ reflection.

备注：内容来源于stack exchange，提问作者Heidi Landon