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A one-to-one function, also known as an injective function, is a kind of function in which every element of the function's domain is paired with a distinct and unique element in its codomain. This characteristic means that no two different inputs will map to the same output. Understanding whether a function is one-to-one is crucial in inverse function problems because only one-to-one functions have inverse functions that are also functions. One-to-one functions can often be recognized by using the horizontal line test.

• If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one.
• If each horizontal line intersects the graph at most once, the function passes the test and is one-to-one.

For example, in the original exercise, our function given is explicitly stated as one-to-one, which reassures us that finding an inverse function is meaningful and possible.

Function composition involves applying one function to the results of another. It is a vital concept when verifying inverse functions. In essence, the composition of a function and its inverse should yield the original input value. Mathematically, this can be expressed as:

• For a function \( f \) and its inverse \( f^{-1} \), the composition \( f(f^{-1}(x)) = x \).
• Similarly, \( f^{-1}(f(x)) = x \) should also hold true.

Checking these compositions is the final step to confirm that two functions are true inverses of each other. In our exercise, once we derived the inverse \( f^{-1}(x) = \frac{x}{1 + 3x} \), we need to verify it by checking these compositions to ensure correctness.

Algebraic manipulation is a process that involves rearranging and simplifying equations to find the solution or solve for a variable of interest. When finding inverse functions, this skill is crucial. In the exercise provided, we start by setting \( y = f(x) = \frac{x}{1 - 3x} \). From here, various algebraic manipulations are performed to isolate \( x \) in terms of \( y \):

• Initial equation: \( y = \frac{x}{1 - 3x} \).
• Cross-multiply to eliminate fractions, which we will explore more in the next section.
• Rearrange and factor out \( x \) from the relevant terms to solve for it.

Every step involves careful application of mathematical principles like distributing terms, combining like terms, and factoring to gradually unravel \( x \) as a function of \( y \). This ultimately bridges the way to finding \( f^{-1}(x) \).

Cross-multiplication is a technique used to eliminate fractions by multiplying the numerator of one fraction by the denominator of another, and vice-versa. This technique is especially useful when dealing with equations where variables are in fractions, as it allows you to work with simpler, fraction-free equations. In the original problem:

• We start with the equation \( y = \frac{x}{1 - 3x} \).
• Cross-multiplying gives: \( y(1 - 3x) = x \).
• This results in the simpler equation \( y - 3yx = x \), which can be further manipulated.

By applying cross-multiplication early in the problem, it simplifies subsequent algebraic manipulation steps, making it more straightforward to solve for \( x \) in terms of \( y \). This step is a cornerstone in solving for inverse functions when dealing with fractional forms.