Find The Value Of The Constant $m$ That Makes The Following Function Continuous On $(-\infty, \infty)$.$f(x) = \begin{cases} mx - 12 & \text{if } X \ \textless \ -4 \\ x^2 + 7x - 8 & \text{if } X \geq -4

Introduction

In mathematics, a function is considered continuous if it can be drawn without lifting the pencil from the paper. In other words, a function is continuous if its graph has no gaps or jumps. In this article, we will discuss how to find the value of the constant $m$ that makes the given function continuous on the interval $(-\infty, \infty)$.

The Function

The given function is defined as:

$$f(x) = \begin{cases} mx - 12 & \text{if } x \ \textless \ -4 \\ x^2 + 7x - 8 & \text{if } x \geq -4 \end{cases}$$

This function has two parts: one for $x < -4$ and another for $x \geq -4$. To find the value of $m$ that makes the function continuous, we need to ensure that the two parts of the function meet at the point $x = -4$.

Continuity at x = -4

For the function to be continuous at $x = -4$, the following conditions must be met:

1. The function must be defined at $x = -4$.
2. The limit of the function as $x$ approaches $-4$ from the left must be equal to the limit of the function as $x$ approaches $-4$ from the right.
3. The limit of the function as $x$ approaches $-4$ must be equal to the value of the function at $x = -4$.

Condition 1: Function Defined at x = -4

The function is defined at $x = -4$ if the value of the function at $x = -4$ is finite. In this case, the value of the function at $x = -4$ is:

$$f(-4) = (-4)^2 + 7(-4) - 8 = 16 - 28 - 8 = -20$$

Since the value of the function at $x = -4$ is finite, the function is defined at $x = -4$.

Condition 2: Limits from the Left and Right

To find the limit of the function as $x$ approaches $-4$ from the left, we use the part of the function defined for $x < -4$:

$$\lim_{x \to -4^-} f(x) = \lim_{x \to -4^-} (mx - 12)$$

To find the limit of the function as $x$ approaches $-4$ from the right, we use the part of the function defined for $x \geq -4$:

$$\lim_{x \to -4^+} f(x) = \lim_{x \to -4^+} (x^2 + 7x - 8)$$

Limit from the Left

To find the limit of the function as $x$ approaches $-4$ from the left, we can substitute $x = -4$ into the expression:

$$\lim_{x \to -4^-} f(x) = m(-4) - 12 = -4m - 12$$

Limit from the Right

To find the limit of the function as $x$ approaches $-4$ from the right, we can substitute $x = -4$ into the expression:

$$\lim_{x \to -4^+} f(x) = (-4)^2 + 7(-4) - 8 = 16 - 28 - 8 = -20$$

Equating the Limits

Since the function is continuous at $x = -4$, the limit of the function as $x$ approaches $-4$ from the left must be equal to the limit of the function as $x$ approaches $-4$ from the right:

$$-4m - 12 = -20$$

Solving for m

To solve for $m$, we can add $12$ to both sides of the equation:

$$-4m = -8$$

Then, we can divide both sides of the equation by $-4$:

$$m = \frac{-8}{-4} = 2$$

Therefore, the value of $m$ that makes the function continuous on the interval $(-\infty, \infty)$ is $m = 2$.

Conclusion

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Introduction

In our previous article, we discussed how to find the value of the constant $m$ that makes the given function continuous on the interval $(-\infty, \infty)$. In this article, we will answer some frequently asked questions related to the topic.

Q: What is continuity in mathematics?

A: Continuity in mathematics refers to the property of a function that can be drawn without lifting the pencil from the paper. In other words, a function is continuous if its graph has no gaps or jumps.

Q: Why is continuity important in mathematics?

A: Continuity is important in mathematics because it allows us to analyze and understand the behavior of functions. Continuous functions can be used to model real-world phenomena, such as population growth, temperature changes, and economic trends.

Q: What are the conditions for continuity at a point?

A: The conditions for continuity at a point are:

1. The function must be defined at the point.
2. The limit of the function as $x$ approaches the point from the left must be equal to the limit of the function as $x$ approaches the point from the right.
3. The limit of the function as $x$ approaches the point must be equal to the value of the function at the point.

Q: How do I find the limit of a function as $x$ approaches a point from the left?

A: To find the limit of a function as $x$ approaches a point from the left, you can use the part of the function defined for $x < a$, where $a$ is the point. You can substitute $x = a$ into the expression and simplify.

Q: How do I find the limit of a function as $x$ approaches a point from the right?

A: To find the limit of a function as $x$ approaches a point from the right, you can use the part of the function defined for $x \geq a$, where $a$ is the point. You can substitute $x = a$ into the expression and simplify.

Q: What is the value of $m$ that makes the function continuous on the interval $(-\infty, \infty)$?

A: The value of $m$ that makes the function continuous on the interval $(-\infty, \infty)$ is $m = 2$.

Q: Can I use the same method to find the value of $m$ for a different function?

A: Yes, you can use the same method to find the value of $m$ for a different function. However, you will need to modify the method to suit the specific function.

Q: What are some real-world applications of continuous functions?

A: Some real-world applications of continuous functions include:

• Modeling population growth
• Analyzing temperature changes
• Studying economic trends
• Predicting stock prices
• Understanding the behavior of physical systems

Conclusion

In this article, we answered some frequently asked questions related to finding the value of $m$ for a continuous function. We discussed the importance of continuity in mathematics, the conditions for continuity at a point, and how to find the limit of a function as $x$ approaches a point from the left and right. We also provided some real-world applications of continuous functions.