Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$\left(\frac{g}{f}\right)(0)$$

Step-by-Step Solution

Answer

The expression is undefined because it involves division by zero.

Step 1: Understand the expression

The expression \(\left(\frac{g}{f}\right)(x)\) means we need to find the function \(\frac{g(x)}{f(x)}\) and then evaluate it at \(x = 0\).

Step 2: Find \(g(0)\) and \(f(0)\)

Evaluate \(g(x) = 2x - 1\) at \(x = 0\): \(g(0) = 2(0) - 1 = -1\). Evaluate \(f(x) = x^2 + 3x\) at \(x = 0\): \(f(0) = 0^2 + 3(0) = 0\).

Step 3: Substitute values into the composition

Substitute \(g(0) = -1\) and \(f(0) = 0\) into the expression \(\frac{g}{f}(0) = \frac{g(0)}{f(0)}\) to get \(\frac{-1}{0}\).

Step 4: Evaluate the result

The expression \(\frac{-1}{0}\) is undefined because division by zero is not allowed in mathematics.

Key Concepts

Evaluating Functions

When working with functions such as \(f(x)\) and \(g(x)\), evaluating them is a key task in understanding their behavior. Evaluating a function simply means finding its output for a particular input value. For instance, if we have \(g(x) = 2x - 1\) and need to evaluate it at \(x = 0\), we substitute 0 into the function:

• Replace every instance of \(x\) in \(g(x)\) with 0.
• Thus, \(g(0) = 2(0) - 1\).
• Simplify to find \(g(0) = -1\).

Next, consider the function \(f(x) = x^2 + 3x\). To evaluate it at \(x = 0\), we follow similar steps:

• Substitute 0 into \(f(x)\) to get \(f(0) = 0^2 + 3(0)\).
• This results in \(f(0) = 0\).

By evaluating functions at specific values, you ascertain specific outputs of these functions, which is fundamentally useful when composing functions.

Division by Zero

One of the most important rules in mathematics is that division by zero is undefined. This is often a source of confusion but can be better understood with some fundamental insights. Imagine dividing a cake (numerator) among zero people (denominator): how can this division happen? It doesn't make sense to distribute the cake when there's no one to receive it. In mathematical terms:

• Any expression of form \(\frac{a}{0}\), where \(a\) is a non-zero number, is undefined.
• This is because multiplying zero by any number results in zero, making it impossible to reverse the operation through division.

In our exercise, you encounter \(\frac{-1}{0}\). Since you cannot divide \(-1\) by zero, the result is an undefined expression. This principle underscores the importance of ensuring that the denominator is always non-zero when performing division.

Undefined Expression

An undefined expression arises when the conditions required for a mathematical operation are not met. Division by zero is a classic example, producing an expression that isn't defined in the real number system. When you evaluate an expression and encounter an undefined result, it indicates a breakdown in normal mathematical operations. Here's how to approach such scenarios:

• First, check the denominator, as division is the most common cause of undefined expressions.
• If the denominator is zero, recognize the expression as undefined.

In our exercise, when substituting into \(\frac{g}{f}(0) = \frac{g(0)}{f(0)}\), you obtain \(\frac{-1}{0}\), an undefined expression. Recognizing undefined expressions is crucial in evaluating compositions of functions or solving equations, ensuring results are meaningful and logical.