Is 1/x Injective?

In the realm of mathematics, the concept of injectivity is fundamental. A function $f:A\to B$f:A→B is said to be injective (or one-to-one) if different inputs produce different outputs. In other words, $f\left(x_1\right)=f\left(x_2\right)$f(x1​)=f(x2​) implies that $x_1=x_2$x1​=x2​. For the function $f\left(x\right)=\frac{1}{x}$f(x)=x1​, this property is particularly interesting to examine.

At first glance, it might seem straightforward to determine whether $\frac{1}{x}$x1​ is injective. Let's dive into the details and see why this function holds the property of injectivity, and also discuss its domain and range to fully understand its behavior.

Domain and Range: The function $\frac{1}{x}$x1​ is defined for all $x\ne 0$x=0. This is because division by zero is undefined. The range of $\frac{1}{x}$x1​ is all real numbers except zero, since $\frac{1}{x}$x1​ can take any real value depending on $x$x.

Proof of Injectivity: To prove that $\frac{1}{x}$x1​ is injective, we need to show that if $\frac{1}{x_1}=\frac{1}{x_2}$x1​1​=x2​1​, then $x_1=x_2$x1​=x2​.

Starting with the equation: $\frac{1}{x_1}=\frac{1}{x_2}$x1​1​=x2​1​

We can cross-multiply to eliminate the fractions: $1\cdot x_2=1\cdot x_1$1⋅x2​=1⋅x1​ $x_2=x_1$x2​=x1​

This demonstrates that if two inputs produce the same output, then those inputs must be identical. Thus, $\frac{1}{x}$x1​ is indeed injective.

Graphical Insight: A graphical representation of $f\left(x\right)=\frac{1}{x}$f(x)=x1​ further supports this finding. The graph of $\frac{1}{x}$x1​ is a hyperbola with two branches, one in each of the first and third quadrants. It approaches the x-axis and y-axis asymptotically, but never touches them. Each value of $x$x maps to a unique $y$y, reinforcing the injective property.

Limitations and Considerations: While $\frac{1}{x}$x1​ is injective, it's worth noting its limitations. The function is not defined at $x=0$x=0, and the behavior near zero should be understood in the context of limits. Additionally, the function does not cover all possible y-values; specifically, it never reaches zero.

Conclusion: The function $\frac{1}{x}$x1​ is a clear example of an injective function. It maps distinct inputs to distinct outputs, as demonstrated both algebraically and graphically. This property makes it useful in various mathematical contexts, including calculus and algebra, where one-to-one relationships are crucial.

By analyzing the function’s injectivity, we gain insights into its behavior and applications, showcasing the elegance and utility of mathematical functions in understanding and solving complex problems.