Given that $g(x) = \frac{x - 1}{x + 5}$, find each of the following:
Final answers: a) \(\boxed{\frac{1}{7}}\), b) \(\boxed{0}\), c) \(\boxed{\text{undefined}}\), d) \(\boxed{\frac{59}{35}}\), e) \(\boxed{\frac{x+h-1}{x+h+5}}\).
The given function is defined as \(g(x) = \frac{x-1}{x+5}\). This is a rational function where the numerator is \(x-1\) and the denominator is \(x+5\). The function is defined for all real values of \(x\) except for \(x = -5\), where the denominator becomes zero, rendering the function undefined at that point. We will evaluate this function for the requested inputs by direct substitution.
To find \(g(2)\), we substitute \(x = 2\) into the function expression:
\[g(2) = \frac{2-1}{2+5} = \frac{1}{7}.\]
So, the result of \(g(2)\) is \(\frac{1}{7}\).
To find \(g(1)\), we substitute \(x = 1\) into the function expression:
\[g(1) = \frac{1-1}{1+5} = \frac{0}{6} = 0.\]
So, the result of \(g(1)\) is \(0\).
To find \(g(-5)\), we examine the denominator:
\[g(-5) = \frac{-5-1}{-5+5} = \frac{-6}{0}.\]
Since division by zero is undefined in real arithmetic, the function \(g(x)\) is undefined at \(x = -5\).
To find \(g(-13.75)\), we substitute \(x = -13.75\) into the function expression:
\[g(-13.75) = \frac{-13.75 - 1}{-13.75 + 5} = \frac{-14.75}{-8.75}.\]
To simplify \(\frac{-14.75}{-8.75}\), we can multiply both the numerator and denominator by 100 to remove the decimals: \(\frac{1475}{875}\). Dividing both terms by their greatest common divisor, 25, we obtain:
\[\frac{1475 \div 25}{875 \div 25} = \frac{59}{35}.\]
So, the result of \(g(-13.75)\) is \(\frac{59}{35}\).
To find \(g(x+h)\), we replace every occurrence of \(x\) with \((x+h)\) in the original expression:
\[g(x+h) = \frac{(x+h)-1}{(x+h)+5} = \frac{x+h-1}{x+h+5}.\]
This expression is already in its simplest form as there are no common factors to cancel between the numerator and the denominator. Thus, \(g(x+h) = \frac{x+h-1}{x+h+5}\).
Rational functions are widely used in economics and finance to model relationships where output or cost is dependent on input. For instance, a simple cost-benefit model might use a function like \(g(x) = \frac{x-1}{x+5}\) to represent the efficiency of a production process, where \(x\) represents the investment in resources. Evaluating this at different points allows managers to predict the efficiency level at different levels of funding, and identifying undefined points (like \(x = -5\)) helps avoid scenarios where the model breaks down, such as when costs are negative. Understanding these functions helps in making data-driven decisions in business and engineering.