Problem 11 - Essential Calculus Early Transcendentals

Question

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. $$ g(x)=1 / x $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function given is \( g(x) = \frac{1}{x} \). It is a rational function that is defined for all real numbers except \( x = 0 \).

Step 2: Definition of a One-to-One Function

A function is one-to-one if different inputs lead to different outputs. In other words, for every \( a eq b \), \( g(a) eq g(b) \). Alternatively, each output is the result of exactly one input.

Step 3: Apply the Horizontal Line Test

Graphically, a function is one-to-one if every horizontal line intersects the graph at most once. Sketch the graph of \( g(x) = \frac{1}{x} \). The graph is a hyperbola with branches in the first and third quadrants. No horizontal line will cross it more than once.

Step 4: Algebraic Verification

To verify algebraically, assume \( g(a) = g(b) \) and check if \( a = b \). For \( g(x) = \frac{1}{x} \), if \( \frac{1}{a} = \frac{1}{b} \), then we can cross-multiply to get \( b = a \). This confirms that no two different \( x \)-values map to the same \( y \).

Step 5: Conclusion about One-to-Oneness

Since each horizontal line intersects the graph at most once (verified graphically) and the function \( g(x) = \frac{1}{x} \) satisfies the algebraic condition for a one-to-one function, \( g(x) = \frac{1}{x} \) is a one-to-one function.

Key Concepts

Rational Function

Rational functions are an important class of functions in mathematics. A rational function is expressed as the quotient of two polynomials. The general form is:

• \[g(x) = \frac{p(x)}{q(x)}\]
• where both \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \).

In the case of \( g(x) = \frac{1}{x} \), it is a simple rational function where \( p(x) = 1 \) and \( q(x) = x \). The function is defined for all real numbers except where the denominator equals zero, in this case, \( x eq 0 \).

Rational functions typically have vertical asymptotes where the denominator is zero and horizontal asymptotes that can be determined by examining the degrees of \( p(x) \) and \( q(x) \). In our example, \( g(x) \) has a vertical asymptote at \( x = 0 \) and approaches zero as \( x \) moves away from the origin. Understanding these characteristics helps in sketching and analyzing these functions.

Horizontal Line Test

The horizontal line test is a visual way to determine whether a given function is one-to-one.

• A function is considered one-to-one if no horizontal line intersects its graph more than once.

Applying this to \( g(x) = \frac{1}{x} \), you would sketch the graph, which forms a hyperbola extending into the first and third quadrants. The key is that horizontal lines passing through this graph will only touch it at one point.

This is because each value in the range of the function corresponds to exactly one value in the domain. Hence, the horizontal line test affirms that \( g(x) = \frac{1}{x} \) is indeed one-to-one. Practicing this test with various functions helps solidify understanding of function behaviors.

Algebraic Verification

Algebraic verification is another method to check that a function is one-to-one. This method involves analyzing the functional equation directly. For \( g(x) = \frac{1}{x} \), suppose:

• \( g(a) = g(b) \)
• Then, \( \frac{1}{a} = \frac{1}{b} \)

By cross-multiplying, you obtain \( a \times 1 = b \times 1 \), simplifying to \( a = b \).

This verification confirms that if two outputs of the function are equal, their inputs must have been equal too.

Thus, through this algebraic approach, we also confirm the one-to-oneness of the function \( g(x) = \frac{1}{x} \). The process of algebraic verification is valuable in providing a non-visual proof of function characteristics, reinforcing the conclusions drawn from graphical methods like the horizontal line test.