Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=3+\frac{2}{x-1}\)

Step-by-Step Solution

Answer

The real value of \(x\) for which \(f(x)=0\) is \(x = \frac{1}{3}\)

Step 1: Set the function to zero

Set the function equal to zero: \(0 = 3 + \frac{2}{x-1}\)

Step 2: Solve for \(x\)

To find the value of \(x\) that satisfies this equation, first solve the equation in the form \(\frac{2}{x-1} = -3\). Cross-multiply to get \(2 = -3(x-1)\), which simplifies as \(2 = -3x + 3\). Reordering the equation gives \(3x = 1\). Finally, divide both sides by 3 to solve for \(x\), which gives us \(x = \frac{1}{3}\).

Step 3: Check for validity

It’s crucial to check the validity of the obtained \(x\) value as there might be a chance it makes the denominator zero, causing division by zero. In our case, \(x = \frac{1}{3}\) does not cause the denominator to become zero, therefore the solution is valid.

Key Concepts

Rational Functions

Rational functions are expressions that involve fractions where both the numerator and the denominator are polynomial expressions. A general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials.
These functions can exhibit characteristics such as vertical asymptotes and discontinuities. Understanding how to manipulate and solve rational functions is a fundamental skill in algebra.
When working with rational functions, the denominator plays a critical role due to the potential for undefined values. In any expression \( \frac{A}{B} \), if \( B \) equates to zero, the expression becomes undefined, which needs careful consideration when solving relevant equations.

Zero of a Function

The zero of a function, also called the root, is the value of \( x \) that makes the function equal to zero. In other words, it's the point where the graph of the function intersects the x-axis.
To find a function's zero, set \( f(x) = 0 \) and solve for \( x \).
In the given exercise, we set \( 0 = 3 + \frac{2}{x-1} \). By moving the constant across the equation, you rearrange this to isolate our rational expression on one side, forming \( \frac{2}{x-1} = -3 \). Solving this equation for \( x \) gives us the zero of the function.
Finding these zeros is essential for sketching graphs, determining intercepts, and understanding the behavior of functions.

Division by Zero

Division by zero is a primary concern in rational functions because it leads to undefined expressions. Mathematically, division by zero does not make sense because there's no number that multiplied by zero gives a non-zero result.
In the equation \( \frac{A}{B} \), if \( B = 0 \), the expression is undefined, leading to potential asymptotes or discontinuities in the function's graph.To prevent division by zero while solving rational functions, always check the denominator's value after finding a potential solution. For instance, when \( x = 1 \) in the original problem, the denominator \( x-1 \) equals zero. Any such solution needs to be excluded or handled with care.
Avoidance of division by zero ensures the calculated values do not lead to undefined conditions, preserving the integrity of any algebraic solution or analysis.