- Define the function \(\left(\frac{f}{g}\right)(x) = \frac{2x+3}{-2x^2-3x}\).
- Factor the denominator: \(-x(2x+3)\).
- Simplify the expression: \(-\frac{1}{x}\) for \(x
eq -\frac{3}{2}\).
- Exclude values where the original denominator is zero: \(x
eq -\frac{3}{2}\) and \(x
eq 0\).
- Final expression: \(\boxed{-\frac{1}{x}}\), \(x
eq \boxed{-\frac{3}{2}}\) or \(\boxed{0}\)
Explanation
- Problem Analysis
We are given the functions \(f(x) = 2x + 3\) and \(g(x) = -2x^2 - 3x\). The objective is to compute the rational function \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\) and determine the values of \(x\) for which this expression is undefined. The function is undefined when the denominator \(g(x) = 0\).
- Simplifying the Expression
The expression \(\left(\frac{f}{g}\right)(x)\) is formed by the ratio:
\[\left(\frac{f}{g}\right)(x) = \frac{2x + 3}{-2x^2 - 3x}\]
To simplify this rational expression, we factor the denominator. The expression \(-2x^2 - 3x\) has a common factor of \(-x\), which allows us to rewrite it as:
\[-2x^2 - 3x = -x(2x + 3)\]
Thus, the expression becomes:
\[\left(\frac{f}{g}\right)(x) = \frac{2x + 3}{-x(2x + 3)}\]
Assuming \(2x + 3
eq 0\), we can cancel the common factor \((2x + 3)\) from the numerator and the denominator:
\[\left(\frac{f}{g}\right)(x) = \frac{1}{-x} = -\frac{1}{x}\]
This simplification is valid provided that \(2x + 3
eq 0\), which implies \(x
eq -\frac{3}{2}\).
- Identifying Excluded Values and Conclusion
The rational function \(\left(\frac{f}{g}\right)(x)\) is undefined whenever the original denominator \(g(x) = 0\). We set the factored form of the denominator to zero to find the excluded values:
\[-x(2x + 3) = 0\]
This equation holds true if \(x = 0\) or \(2x + 3 = 0\). Solving for \(x\) in the second case gives \(x = -\frac{3}{2}\). Therefore, the function is undefined at \(x = 0\) and \(x = -\frac{3}{2}\). The simplified expression is \(-\frac{1}{x}\), where \(x
eq -\frac{3}{2}\) and \(x
eq 0\).
Examples
Rational functions are essential in engineering and physics, often used to model relationships where one variable depends on the ratio of two others. For example, in electrical circuits, the total resistance of resistors connected in parallel involves the reciprocal of the sum of individual reciprocal resistances. Simplifying expressions like this allows engineers to quickly determine how a system's behavior changes as certain input parameters (represented by \(x\)) approach specific critical values or limits. Mastering this algebraic technique ensures accurate modeling and prevents division by zero errors in complex design calculations.