Problem 29 At what points are the functions... [FREE SOLUTION]

Chapter 2: Problem 29

At what points are the functions in Exercises $13-30$ continuous? $$g(x)=\left\\{\begin{array}{ll} \frac{x^{2}-x-6}{x-3}, & x \neq 3 \\ 5, & x=3 \end{array}\right.$$

Short Answer

Expert verified

The function is continuous for all $x$, since the limit at $x=3$ matches the function value.

Step by step solution

Review the Function

The function $g(x)$ is a piecewise function defined as $\frac{x^2-x-6}{x-3}$ for $x eq 3$ and as 5 when $x=3$. To determine the points of continuity, we need to analyze the behavior of the function at different values of $x$, particularly around $x=3$.

Simplify the Function for $x \neq 3$

The expression $\frac{x^2-x-6}{x-3}$ can be simplified. Factoring the numerator, we have $(x-3)(x+2)$, so $\frac{(x-3)(x+2)}{x-3}$. For $x eq 3$, this simplifies to $x+2$.

Check Continuity at $x=3$

For a function to be continuous at a point, the limit as $x$ approaches that point must exist and equal the value of the function at that point. Thus, we need $\lim_{x \to 3} g(x) = g(3) = 5$.

Evaluate the Limit as $x \to 3$

Substitute the simplified form of the function for $x eq 3$: $\lim_{x \to 3} (x+2) = 3 + 2 = 5$. The limit exists and equals 5.

Compare the Limit with $g(3)$

Since $g(3) = 5$, and $\lim_{x \to 3} g(x) = 5$, the function meets the condition for continuity at $x=3$.

Continuity at Other Points

For $x eq 3$, $g(x) = x+2$, which is a polynomial function. Polynomial functions are continuous at all points, so $g(x)$ is continuous for all $x eq 3$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Piecewise Functions

A piecewise function is a mathematical expression defined by different formulas or expressions over different parts of its domain. They allow us to describe a function that may behave differently in separate intervals:

The given function, $ g(x) $, is defined differently for $ x = 3 $ and $ x eq 3 $.
For $ x = 3 $, $ g(x) = 5 $.
For $ x eq 3 $, the function is expressed as $ \frac{x^2 - x - 6}{x - 3} $.

Piecewise functions require special attention when discussing continuity, as each section of the function could introduce discontinuities. Understanding how to handle the points where the function changes is crucial to solving such exercises.

Limit of a Function

The limit of a function describes the behavior of the function as it gets infinitely close to a specific point. It's a foundational concept in calculus, used to understand continuity:

To check if $ g(x) $ is continuous at $ x = 3 $, we find the limit of $ g(x) $ as $ x $ approaches 3.
If $ \lim_{x \to 3} g(x) = g(3) $, then $ g(x) $ is continuous at $ x = 3 $.

For the function given, this means evaluating the limit of the simplified form $ x + 2 $ as $ x $ approaches 3. Because this equals 5, and $ g(3) = 5 $, the function is continuous at this point. Limits help determine whether a function "fits together" smoothly at a given point.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler "factor" polynomials whose product is the original polynomial. It鈥檚 a critical step in simplifying expressions:

The numerator $ x^2 - x - 6 $ factors into $ (x - 3)(x + 2) $.
This breakdown allows us to simplify $ \frac{x^2 - x - 6}{x - 3} $ to $ x + 2 $ when $ x eq 3 $.

This simplification is vital, as it removes the apparent discontinuity at $ x = 3 $ and reveals the underlying behavior of the function away from that point. Factoring ensures that we capture the true nature of the function, excluding artificial restrictions.

Polynomial Functions

Polynomial functions are algebraic expressions consisting of terms in the form $ ax^n $, where $ n $ is a non-negative integer. Such functions are significant in calculus for their continuity characteristics:

The expression $ x + 2 $ is a polynomial function of degree 1.
Polynomials are continuous for all real numbers, meaning $ g(x) = x + 2 $ is continuous for $ x eq 3 $.

The continuity of polynomial functions simplifies checking the points of continuity in piecewise functions. When one section of a piecewise function is a polynomial, as here for $ x eq 3 $, it simplifies the overall continuity analysis.

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91影视

At what points are the functions in Exercises \(13-30\) continuous? $$g(x)=\left\\{\begin{array}{ll} \frac{x^{2}-x-6}{x-3}, & x \neq 3 \\ 5, & x=3 \end{array}\right.$$

Short Answer

Step by step solution

Review the Function

Simplify the Function for \(x \neq 3\)

Check Continuity at \(x=3\)

Evaluate the Limit as \(x \to 3\)

Compare the Limit with \(g(3)\)

Continuity at Other Points

Key Concepts

Piecewise Functions

Limit of a Function

Factoring Polynomials

Polynomial Functions

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