Problem 36 - Precalculus

Question

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$g(x)=10 x-x^{2} ; x \rightarrow-2$$

Step-by-Step Solution

Answer

As \(x\) approaches \(-2\), \(g(x)\) approaches \(-24\): \(\lim_{{x \to -2}} g(x) = -24\).

Step 1: Setting Up the Table

To evaluate the function \(g(x) = 10x - x^2\) as \(x\) approaches \(-2\), we will create a table of values for \(x\) around \(-2\). Choose values slightly less than \(-2\), slightly more than \(-2\), and equal to \(-2\). Values such as \(-2.1\), \(-2\), and \(-1.9\) are good starting points.

Step 2: Calculating \(g(x)\) Values

Calculate \(g(x)\) for each chosen \(x\).1. For \(x = -2.1\): \(g(-2.1) = 10(-2.1) - (-2.1)^2 = -21 - 4.41 = -25.41\) 2. For \(x = -2\): \(g(-2) = 10(-2) - (-2)^2 = -20 - 4 = -24\) 3. For \(x = -1.9\): \(g(-1.9) = 10(-1.9) - (-1.9)^2 = -19 - 3.61 = -22.61\)

Step 3: Analyzing \(g(x)\) Values

By examining the values calculated:- As \(x\) approaches \(-2\), \(g(x)\) seems to approach \(-24\) since \(g(-2) = -24\), and the calculated values move towards that as \(x\) nears \(-2\).

Step 4: Writing the Limit Notation

Since \(g(x)\) approaches \(-24\) as \(x\) approaches \(-2\), we can write: \[\lim_{{x \to -2}} g(x) = -24\]In words, as \(x\) nears \(-2\), the function \(g(x)\) gets closer to \(-24\).

Key Concepts

Evaluating Functions

Evaluating functions is a core aspect of understanding how a mathematical relationship behaves over different inputs. In this exercise, we were given the function \( g(x) = 10x - x^2 \), which is a quadratic function. This means that the function maps each \( x \) value in the domain to a unique \( g(x) \) output based on the equation provided. To evaluate this function at specific points, you substitute the specific \( x \) values directly into the function. For instance, when substituting \( x = -2 \) into our function, you calculate \( g(-2) = 10(-2) - (-2)^2 = -20 - 4 = -24 \). By repeating this process for values near \( x = -2 \), you can predict how the function behaves around that point. This is essential for understanding concepts such as continuity and approaching limits in precalculus.Evaluating functions involves:

• Inserting chosen \( x \) values into the function equation.
• Performing arithmetic operations as defined by the function rule.
• Interpreting the result to understand the function's behavior at and near specific points.

Table of Values

When trying to evaluate a function as \( x \) approaches a particular value, using a table is a powerful visual tool. It helps organize the data for each \( x \) value you choose to evaluate and provides a clear picture of how the function behaves around that point.Creating a table of values involves choosing \( x \) values near the point of interest—in this case, \( -2 \). You then calculate the corresponding \( g(x) \) values based on the function. For instance:

• For \( x = -2.1 \), find \( g(x) \).
• For \( x = -2 \), find \( g(x) \).
• For \( x = -1.9 \), find \( g(x) \).

Using these calculations, the table outlines a clear trend, helping to visualize whether \( g(x) \) approaches a particular value. This method simplifies the process of identifying patterns and is particularly useful in contexts without graphical aids.

Limit Notation

Limit notation is a concise way to express the behavior of a function as its input (\( x \)) approaches a specific value. In this exercise, as we observed that \( g(x) \) approaches \(-24\) as \( x \) approaches \(-2\), limit notation allows us to describe this behavior mathematically as \( \lim_{{x \to -2}} g(x) = -24 \).Understanding limit notation involves:

• The limit symbol \( \lim \) which signifies approaching behavior.
• The subscript \( x \to -2 \), showing the input approaching the value -2.
• The function \( g(x) \) on which the limit operates.
• The resulting limit value of \(-24\) which the function approaches.

In words, this notation is expressed as: "As \( x \) nears \(-2\), the function \( g(x) \) tends toward \(-24\)." Limit notation is fundamental in precalculus to communicate these ideas succinctly and is foundational for concepts explored in calculus.