Animated Solution for Mathematics - Functions: Let $f:(0,1)\to R$ be defined by $f(x)=\frac{b-x}{1-bx}$, where $b$ is a constant such that $0<b<1$. Then

Imagine you are an engineer designing a machine that takes an input $x$ and transforms it into an output defined by the function:

You are told that $b$ is a constant, a fixed parameter such that $0<b<1$. To determine if this machine is a perfect mirror—that is, if it is invertible—we must analyze its mathematical properties.

The Monotonicity Check

To determine if the function is one-to-one, we examine its derivative. We apply the quotient rule, where $u=b-x$ and $v=1-bx$. The derivative formula is:

Substituting our values, we obtain:

Expanding the numerator, we get $-1+bx+b^2-bx$. The $bx$ terms cancel out, leaving us with the elegant expression:

Because $0<b<1$, we know $b^2<1$, which implies that $b^2-1$ is strictly negative. Since the denominator is a square, it is always positive. Thus, $f^{\prime }(x)<0$ for all $x$ in the domain, confirming that the function is strictly decreasing and therefore one-to-one.

The Range Trap

A function is invertible only if it is both one-to-one and onto. We must now determine if the range of $f(x)$ covers the entire codomain, $\mathbb{R}$.

Given that the function is strictly decreasing, we examine the behavior at the boundaries of the domain $(0,1)$:

The range of the function is the open interval $(-1,b)$. Since $(-1, b) eq \mathbb{R}$, the function is not onto. Consequently, the machine fails the test for global invertibility.

The Inverse Paradox

Although the function is not invertible on $\mathbb{R}$, we can derive its inverse relation by solving $y=\frac{b-x}{1-bx}$ for $x$:

The algebraic rule for the inverse is identical to the original function. However, we must note that the domain of $f$ is $(0,1)$, while the domain of $f^{-1}$ is $(-1,b)$.

Because these domains differ, $f$ and $f^{-1}$ are not the same function. You have successfully navigated the traps of this problem by respecting the strict definitions of functions, domains, and ranges.