Slope at the Point of Discontinuity

Oct 13, 2025 by windailys.com 19 views

Introduction

The function $f(x)=3|x+2|-3$ is a linear function with a slope of 3. However, the absolute value function $|x+2|$ introduces a change in the slope at the point where $x+2=0$ , which is $x=-2$ . This change in slope is due to the nature of the absolute value function, which changes the sign of the expression inside the absolute value bars.

Vertex

The vertex of the function $f(x)=3|x+2|-3$ is the point where the function changes from decreasing to increasing or vice versa. To find the vertex, we need to find the point where the function changes from decreasing to increasing.

The function $f(x)=3|x+2|-3$ can be rewritten as:

f(x)=\begin{cases} 3(-x-2)-3 & \text{if } x < -2 \\ 3(x+2)-3 & \text{if } x \geq -2 \end{cases}

For $x < -2$ , the function is $f(x)=-3x-9$ . For $x \geq -2$ , the function is $f(x)=3x+3$ .

The vertex of the function is the point where the two pieces of the function meet, which is at $x=-2$ . To find the y-coordinate of the vertex, we can plug in $x=-2$ into either piece of the function.

Plugging in $x=-2$ into the first piece of the function, we get:

f(-2)=-3(-2)-9=3

Plugging in $x=-2$ into the second piece of the function, we get:

f(-2)=3(-2)+3=-3

Since the two pieces of the function meet at $x=-2$ , the y-coordinate of the vertex is the average of the two y-coordinates, which is $\frac{3+(-3)}{2}=0$ .

Therefore, the vertex of the function $f(x)=3|x+2|-3$ is at the point $(x, y)=(-2, 0)$ .

Domain

The domain of a function is the set of all possible input values for which the function is defined. In the case of the function $f(x)=3|x+2|-3$ , the domain is all real numbers, since the function is defined for all values of $x$ .

Range

The range of a function is the set of all possible output values for which the function is defined. In the case of the function $f(x)=3|x+2|-3$ , the range is all real numbers greater than or equal to $-6$ , since the minimum value of the function is $-6$ .

Slopes

The slopes of the function $f(x)=3|x+2|-3$ are the rates of change of the function at different points. Since the function is a linear function, the slope is constant and is equal to 3.

However, the absolute value function $|x+2|$ introduces a change in the slope at the point where $x+2=0$ , which is $x=-2$ . This change in slope is due to the nature of the absolute value function, which changes the sign of the expression inside the absolute value bars.

To find the slopes of the function, we can take the derivative of the function with respect to $x$ . The derivative of the function is:

f'(x)=\begin{cases} -3 & \text{if } x < -2 \\ 3 & \text{if } x \geq -2 \end{cases}

The derivative of the function is a piecewise function, since the derivative of the absolute value function is a piecewise function.

The derivative of the function is $-3$ for $x < -2$ and $3$ for $x \geq -2$ . This means that the slope of the function is $-3$ for $x < -2$ and $3$ for $x \geq -2$ .

Conclusion

In conclusion, the function $f(x)=3|x+2|-3$ is a linear function with a slope of 3. However, the absolute value function $|x+2|$ introduces a change in the slope at the point where $x+2=0$ , which is $x=-2$ . This change in slope is due to the nature of the absolute value function, which changes the sign of the expression inside the absolute value bars.

The vertex of the function is at the point $(x, y)=(-2, 0)$ , the domain is all real numbers, the range is all real numbers greater than or equal to $-6$ , and the slopes of the function are $-3$ for $x < -2$ and $3$ for $x \geq -2$ .

Slopes of the Function $f(x)=3|x+2|-3$

Slope for $x < -2$

The slope of the function for $x < -2$ is $-3$ . This means that the function is decreasing for $x < -2$ .

Slope for $x \geq -2$

The slope of the function for $x \geq -2$ is $3$ . This means that the function is increasing for $x \geq -2$ .

Slope at the Vertex

The slope of the function at the vertex is $3$ . This means that the function is increasing at the vertex.

Slope at the Point of Discontinuity

The slope of the function at the point of discontinuity is $-3$ . This means that the function is decreasing at the point of discontinuity.

Graph of the Function

The graph of the function $f(x)=3|x+2|-3$ is a V-shaped graph with a vertex at the point $(x, y)=(-2, 0)$ .

The graph of the function is decreasing for $x < -2$ and increasing for $x \geq -2$ . The graph of the function is also continuous for all values of $x$ except at the point $x=-2$ , where the function is discontinuous.

Conclusion

Q: What is the domain of the function $f(x)=3|x+2|-3$ ?

A: The domain of the function $f(x)=3|x+2|-3$ is all real numbers, since the function is defined for all values of $x$ .

Q: What is the range of the function $f(x)=3|x+2|-3$ ?

A: The range of the function $f(x)=3|x+2|-3$ is all real numbers greater than or equal to $-6$ , since the minimum value of the function is $-6$ .

Q: What is the vertex of the function $f(x)=3|x+2|-3$ ?

A: The vertex of the function $f(x)=3|x+2|-3$ is at the point $(x, y)=(-2, 0)$ .

Q: What is the slope of the function $f(x)=3|x+2|-3$ for $x < -2$ ?

A: The slope of the function $f(x)=3|x+2|-3$ for $x < -2$ is $-3$ .

Q: What is the slope of the function $f(x)=3|x+2|-3$ for $x \geq -2$ ?

A: The slope of the function $f(x)=3|x+2|-3$ for $x \geq -2$ is $3$ .

Q: Is the function $f(x)=3|x+2|-3$ continuous for all values of $x$ ?

A: No, the function $f(x)=3|x+2|-3$ is discontinuous at the point $x=-2$ .

Q: What is the graph of the function $f(x)=3|x+2|-3$ ?

A: The graph of the function $f(x)=3|x+2|-3$ is a V-shaped graph with a vertex at the point $(x, y)=(-2, 0)$ .

Q: Is the function $f(x)=3|x+2|-3$ increasing or decreasing for $x < -2$ ?

A: The function $f(x)=3|x+2|-3$ is decreasing for $x < -2$ .

Q: Is the function $f(x)=3|x+2|-3$ increasing or decreasing for $x \geq -2$ ?

A: The function $f(x)=3|x+2|-3$ is increasing for $x \geq -2$ .

Q: What is the minimum value of the function $f(x)=3|x+2|-3$ ?

A: The minimum value of the function $f(x)=3|x+2|-3$ is $-6$ .

Q: What is the maximum value of the function $f(x)=3|x+2|-3$ ?

A: The maximum value of the function $f(x)=3|x+2|-3$ is $\infty$ , since the function is increasing for $x \geq -2$ .

Q: Is the function $f(x)=3|x+2|-3$ a linear function?

A: Yes, the function $f(x)=3|x+2|-3$ is a linear function.

Q: What is the equation of the function $f(x)=3|x+2|-3$ ?

A: The equation of the function $f(x)=3|x+2|-3$ is $f(x)=3|x+2|-3$ .

Q: What is the graph of the function $f(x)=3|x+2|-3$ ?

A: The graph of the function $f(x)=3|x+2|-3$ is a V-shaped graph with a vertex at the point $(x, y)=(-2, 0)$ .

Q: Is the function $f(x)=3|x+2|-3$ continuous for all values of $x$ ?

A: No, the function $f(x)=3|x+2|-3$ is discontinuous at the point $x=-2$ .

Q: What is the slope of the function $f(x)=3|x+2|-3$ at the vertex?

A: The slope of the function $f(x)=3|x+2|-3$ at the vertex is $3$ .

Q: What is the slope of the function $f(x)=3|x+2|-3$ at the point of discontinuity?

A: The slope of the function $f(x)=3|x+2|-3$ at the point of discontinuity is $-3$ .

Q: Is the function $f(x)=3|x+2|-3$ a piecewise function?

A: Yes, the function $f(x)=3|x+2|-3$ is a piecewise function.

Q: What are the two pieces of the function $f(x)=3|x+2|-3$ ?

A: The two pieces of the function $f(x)=3|x+2|-3$ are $f(x)=-3x-9$ for $x < -2$ and $f(x)=3x+3$ for $x \geq -2$ .

Q: What is the equation of the function $f(x)=3|x+2|-3$ for $x < -2$ ?

A: The equation of the function $f(x)=3|x+2|-3$ for $x < -2$ is $f(x)=-3x-9$ .

Q: What is the equation of the function $f(x)=3|x+2|-3$ for $x \geq -2$ ?

A: The equation of the function $f(x)=3|x+2|-3$ for $x \geq -2$ is $f(x)=3x+3$ .

Q: Is the function $f(x)=3|x+2|-3$ a linear function?

A: Yes, the function $f(x)=3|x+2|-3$ is a linear function.

Q: What is the slope of the function $f(x)=3|x+2|-3$ ?

A: The slope of the function $f(x)=3|x+2|-3$ is $3$ .

Q: What is the y-intercept of the function $f(x)=3|x+2|-3$ ?

A: The y-intercept of the function $f(x)=3|x+2|-3$ is $-3$ .

Q: What is the x-intercept of the function $f(x)=3|x+2|-3$ ?

A: The x-intercept of the function $f(x)=3|x+2|-3$ is $-2$ .

Q: Is the function $f(x)=3|x+2|-3$ a quadratic function?

A: No, the function $f(x)=3|x+2|-3$ is not a quadratic function.

Q: What is the degree of the function $f(x)=3|x+2|-3$ ?

A: The degree of the function $f(x)=3|x+2|-3$ is $1$ .

Q: Is the function $f(x)=3|x+2|-3$ a polynomial function?

A: Yes, the function $f(x)=3|x+2|-3$ is a polynomial function.

Q: What is the leading coefficient of the function $f(x)=3|x+2|-3$ ?

A: The leading coefficient of the function $f(x)=3|x+2|-3$ is $3$ .

Q: What is the constant term of the function $f(x)=3|x+2|-3$ ?

A: The constant term of the function $f(x)=3|x+2|-3$ is $-3$ .

Q: Is the function $f(x)=3|x+2|-3$ a rational function?

A: No, the function $f(x)=3|x+2|-3$ is not a rational function.

Q: What is the domain of the function $f(x)=3|x+2|-3$ ?

A: The domain of the function $f(x)=3|x+2|-3$ is all real numbers.

Q: What is the range of the function $f(x)=3|x+2|-3$ ?

A: The range of the function $f(x)=3|x+2|-3$ is all real numbers greater than or equal to $-6$ .

Q: Is the function $f(x)=3|x+2|-3$ a linear function?

A: Yes, the function $f(x)=3|x+2|-3$ is a linear function.

Q: What is the slope of the function $f(x)=3|x+2|-3$ ?

A: The slope of the function $f(x)=3|x+2|-3$ is $3$ .

Q: What is the y-intercept of the function $f(x)=3|x+2|-3$ ?

A: The y-intercept of the function $f(x)=3|x+2|-3$ is $-3$ .

Q: What is the x-intercept of the function $f(x)=3|x+2|-3$ ?

A: The x-intercept of the function $f(x)=3|x+2|-3$ is $-2$ .

Q: Is the function $f(x)=3|x+2|-3$ a quadratic function?

A: No, the function $f(x)=3|x+2|-3$ is not a quadratic function.