Problem 16 - Algebra and Trigonometry

Question

Determine whether the function is one-to-one. \(f(x)=\frac{1}{x}\)

Step-by-Step Solution

Answer

The function \(f(x)=\frac{1}{x}\) is one-to-one.

Step 1: Understand the definition

A function is one-to-one (or injective) if each value in the domain of the function corresponds to a unique value in the range. In mathematical terms, for the function \(f(x)\), if \(f(a) = f(b)\) implies \(a = b\) for any \(a\) and \(b\) in the domain, then the function is one-to-one.

Step 2: Analyze the function analytically

Let's consider the function \(f(x) = \frac{1}{x}\) and suppose \(f(a) = f(b)\). This means \(\frac{1}{a} = \frac{1}{b}\). For two fractions to be equal, their denominators must be equal as long as the numerators are non-zero, which implies that \(a = b\). Hence, \(f(x)\) satisfies the condition for being one-to-one based solely on analysis.

Step 3: Consider the domain restrictions

The function \(f(x) = \frac{1}{x}\) is defined for all real numbers except \(x = 0\). This does not affect the one-to-one designation, but is important in establishing the function's domain, which is all real numbers except zero.

Step 4: Verify using graph or properties

Graphically, the function \(f(x) = \frac{1}{x}\) is a hyperbola that passes through the origin's line (considered with an asymptote at \(x = 0\)). Each \(y\)-value corresponds to exactly one \(x\)-value (and vice versa), excluding \(x = 0\). Additionally, the function passes the "horizontal line test" - no horizontal line intersects the graph more than once, confirming its one-to-one nature.

Key Concepts

Function Analysis

A critical aspect of understanding if a function is one-to-one involves thorough function analysis. In simple terms, this includes checking if each input (from the domain) corresponds to a distinct output (in the range). For the function \(f(x) = \frac{1}{x}\), when we set \(f(a) = f(b)\), we get \(\frac{1}{a} = \frac{1}{b}\).

This equation simplifies to \(a = b\) by multiplying both sides by \(ab\) (assuming neither \(a\) nor \(b\) is zero). This indicates that different inputs produce different outputs, adhering to the definition of a one-to-one function.

It's essential to ensure there are no repeated outputs for different inputs which affirms the uniqueness required for injectivity.

Domain Restrictions

Domain restrictions are crucial in understanding and verifying the one-to-one nature of a function. The function \(f(x) = \frac{1}{x}\) is not defined at \(x = 0\) because division by zero is undefined. Therefore, the domain excludes \(x = 0\) and only comprises all other real numbers.

While this domain restriction does not affect the one-to-one property of the function directly, it sets boundaries within which the function operates. These restrictions prevent the function from becoming not well-defined or causing calculations that do not have real values.

• Domain of \(f(x) = \frac{1}{x}\): all real numbers except zero
• Exclusion of division by zero prevents undefined behavior

Understanding these boundaries is essential for working with functions and ensuring they remain properly defined across all elements of the domain.

Graphical Verification

Graphical verification solidifies our analytical findings by providing a visual confirmation. The graph of \(f(x) = \frac{1}{x}\) forms a hyperbola with two separate branches, avoiding the \(y\)-axis because \(x=0\) is undefined.

This graph demonstrates a key property of one-to-one functions: it passes the "horizontal line test." This means no horizontal line will intersect the graph more than once, confirming that each horizontal line represents a unique output value associated with only one input. Additionally, points on the graph approach, but never touch, the line \(x = 0\), emphasizing the domain restriction.

• Graph is a hyperbola
• Passes the horizontal line test
• Excludes \(x = 0\) due to asymptotic behavior

Observing and understanding these graphical characteristics help reinforce the analytical understanding of the function being one-to-one.