- Start with the quotient definition: \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\).
- Substitute the given polynomials to obtain \(\frac{-x + 3}{-2x^2 - 7x}\).
- Factor the denominator \(-2x^2 - 7x = -x(2x + 7)\) to identify the values where the function is undefined: \(x = 0\) and \(x = -3.5\).
- Present the final simplified rational function: \(\boxed{\frac{x - 3}{2x^2 + 7x}, x
eq 0, x
eq -3.5}\).
Explanation
- Analysis of the Problem
The objective is to find the expression for the quotient function \(\left(\frac{f}{g}\right)(x)\) given the two functions \(f(x) = -x + 3\) and \(g(x) = -2x^2 - 7x\). The quotient function is defined as \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). We must also identify the values of \(x\) for which the expression is undefined, which occurs when the denominator \(g(x) = 0\).
- Setting up the Quotient Function
We substitute the expressions for \(f(x)\) and \(g(x)\) into the quotient formula:
\[ \left(\frac{f}{g}\right)(x) = \frac{-x + 3}{-2x^2 - 7x} \]
This is the required rational function expression. We can express the denominator as \(-x(2x + 7)\) to simplify analysis.
- Identifying the Domain Restrictions
The rational function is undefined when the denominator is equal to zero. Thus, we set \(g(x) = 0\):
\[ -2x^2 - 7x = 0 \]
Factoring out \(-x\), we get:
\[ -x(2x + 7) = 0 \]
This equation holds when \(-x = 0\) or \(2x + 7 = 0\).
Solving for \(x\):
- \(-x = 0 \implies x = 0\)
- \(2x + 7 = 0 \implies 2x = -7 \implies x = -\frac{7}{2} = -3.5\)
Therefore, the function is defined for all real \(x\) except \(x = 0\) and \(x = -3.5\).
- Final Result
Combining the expression for the quotient and the domain restrictions, we obtain:
\[ \left(\frac{f}{g}\right)(x) = \frac{-x + 3}{-2x^2 - 7x}, \text{ for } x
eq 0 \text{ and } x
eq -3.5 \]
This can be written in a cleaner form by multiplying numerator and denominator by \(-1\):
\[ \left(\frac{f}{g}\right)(x) = \frac{x - 3}{2x^2 + 7x}, \text{ where } x
eq 0 \text{ and } x
eq -3.5 \]
The final answer is \(\boxed{\frac{x - 3}{2x^2 + 7x}, x
eq 0, x
eq -3.5}\).
Examples
In real-world applications, rational functions often represent rates of change or efficiency models. For example, if \(f(x) = -x + 3\) represents the net gain of a system and \(g(x) = -2x^2 - 7x\) represents the power consumption, the ratio \(\left(\frac{f}{g}\right)(x)\) would represent the efficiency of the system at a given parameter \(x\). Identifying values where the denominator is zero (like \(x=0\) and \(x=-3.5\)) is crucial in engineering to avoid "division by zero" errors where the system model breaks down and potentially causes hardware failure or computational instability.