Select All The Correct Locations On The Graph.Consider The Given Piecewise Function:$\[ f(x)= \begin{cases} -(3x+7) & \text{for } X \ \textless \ -3 \\ 2x^2-16 & \text{for } -3 \leq X \leq 3 \\ -\left(2^x-10\right) & \text{for } X \

Understanding Piecewise Functions

A piecewise function is a mathematical function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In other words, a piecewise function is a function that is composed of multiple functions, each of which is defined on a specific interval of the domain. The intervals are usually defined by a set of conditions or constraints, and the function is defined differently on each interval.

The Given Piecewise Function

The given piecewise function is:

${ f(x)= \begin{cases} -(3x+7) & \text{for } x \ \textless \ -3 \\ 2x^2-16 & \text{for } -3 \leq x \leq 3 \\ -\left(2^x-10\right) & \text{for } x \ \end{cases} }$

Analyzing the Function

To analyze the function, we need to understand the behavior of each sub-function on its respective interval.

For x < -3

On this interval, the function is defined as:

${ f(x) = -(3x+7) }$

This is a linear function with a negative slope. The graph of this function will be a straight line with a negative slope, passing through the point (-3, 20).

For -3 ≤ x ≤ 3

On this interval, the function is defined as:

${ f(x) = 2x^2-16 }$

This is a quadratic function with a positive leading coefficient. The graph of this function will be a parabola that opens upwards, passing through the points (-3, 20) and (3, -20).

For x > 3

On this interval, the function is defined as:

${ f(x) = -\left(2^x-10\right) }$

This is an exponential function with a negative coefficient. The graph of this function will be an exponential curve that decreases as x increases, passing through the point (3, 0).

Selecting Correct Locations on the Graph

To select the correct locations on the graph, we need to identify the key features of each sub-function on its respective interval.

Key Features of Each Sub-Function

• For x < -3, the function is a linear function with a negative slope, passing through the point (-3, 20).
• For -3 ≤ x ≤ 3, the function is a quadratic function with a positive leading coefficient, passing through the points (-3, 20) and (3, -20).
• For x > 3, the function is an exponential function with a negative coefficient, passing through the point (3, 0).

Conclusion

In conclusion, the given piecewise function is a complex function that is defined by multiple sub-functions, each applied to a specific interval of the domain. To select the correct locations on the graph, we need to identify the key features of each sub-function on its respective interval. By analyzing the behavior of each sub-function, we can determine the correct locations on the graph.

Key Takeaways

• A piecewise function is a mathematical function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
• The given piecewise function is a complex function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
• To select the correct locations on the graph, we need to identify the key features of each sub-function on its respective interval.
• By analyzing the behavior of each sub-function, we can determine the correct locations on the graph.

Final Thoughts

Q: What is a piecewise function?

A: A piecewise function is a mathematical function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: How do I determine the intervals for a piecewise function?

A: The intervals for a piecewise function are usually defined by a set of conditions or constraints. For example, if a function is defined as:

${ f(x)= \begin{cases} -(3x+7) & \text{for } x \ \textless \ -3 \\ 2x^2-16 & \text{for } -3 \leq x \leq 3 \\ -\left(2^x-10\right) & \text{for } x \ \end{cases} }$

The intervals are defined as x < -3, -3 ≤ x ≤ 3, and x > 3.

Q: How do I graph a piecewise function?

A: To graph a piecewise function, you need to graph each sub-function on its respective interval. For example, if a function is defined as:

${ f(x)= \begin{cases} -(3x+7) & \text{for } x \ \textless \ -3 \\ 2x^2-16 & \text{for } -3 \leq x \leq 3 \\ -\left(2^x-10\right) & \text{for } x \ \end{cases} }$

You would graph the linear function -(3x+7) for x < -3, the quadratic function 2x^2-16 for -3 ≤ x ≤ 3, and the exponential function -\left(2^x-10\right) for x > 3.

Q: How do I find the domain of a piecewise function?

A: To find the domain of a piecewise function, you need to find the intervals where the function is defined. For example, if a function is defined as:

${ f(x)= \begin{cases} -(3x+7) & \text{for } x \ \textless \ -3 \\ 2x^2-16 & \text{for } -3 \leq x \leq 3 \\ -\left(2^x-10\right) & \text{for } x \ \end{cases} }$

The domain is x < -3, -3 ≤ x ≤ 3, and x > 3.

Q: How do I find the range of a piecewise function?

A: To find the range of a piecewise function, you need to find the set of all possible output values. For example, if a function is defined as:

${ f(x)= \begin{cases} -(3x+7) & \text{for } x \ \textless \ -3 \\ 2x^2-16 & \text{for } -3 \leq x \leq 3 \\ -\left(2^x-10\right) & \text{for } x \ \end{cases} }$

The range is all real numbers.

Q: Can I simplify a piecewise function?

A: Yes, you can simplify a piecewise function by combining the sub-functions into a single function. For example, if a function is defined as:

${ f(x)= \begin{cases} -(3x+7) & \text{for } x \ \textless \ -3 \\ 2x^2-16 & \text{for } -3 \leq x \leq 3 \\ -\left(2^x-10\right) & \text{for } x \ \end{cases} }$

You can simplify the function by combining the sub-functions into a single function.

Q: Can I use a piecewise function in real-world applications?

A: Yes, piecewise functions can be used in real-world applications such as modeling population growth, predicting stock prices, and analyzing data.

Conclusion

In conclusion, piecewise functions are a powerful tool for modeling complex phenomena in mathematics and real-world applications. By understanding the behavior of each sub-function on its respective interval, we can determine the correct locations on the graph and find the domain and range of the function.