Find $f(-1)$ For This Piecewise-defined Function:$f(x) = \begin{cases} 2x + \frac{4}{3} & \text{if } X \ \textless \ -3 \ -5x + 1 & \text{if } X \geq -3 \end{cases}$Write Your Answer As An Integer Or As A Fraction In Simplest

Understanding Piecewise-Defined Functions

A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In other words, a piecewise-defined function is a function that has different formulas for different parts of its domain. These functions are commonly used in mathematics and are an essential part of many mathematical concepts, including calculus and algebra.

The Given Piecewise-Defined Function

The given piecewise-defined function is:

$$f(x) = \begin{cases} 2x + \frac{4}{3} & \text{if } x \ \textless \ -3 \\ -5x + 1 & \text{if } x \geq -3 \end{cases}$$

This function has two sub-functions: one for the interval $x < -3$ and another for the interval $x \geq -3$. To find the value of $f(-1)$, we need to determine which sub-function to use.

Determining the Sub-Function to Use

Since $-1 < -3$, we will use the first sub-function: $2x + \frac{4}{3}$.

Finding the Value of $f(-1)$

To find the value of $f(-1)$, we substitute $x = -1$ into the first sub-function:

$$f(-1) = 2(-1) + \frac{4}{3}$$

Simplifying the expression, we get:

$$f(-1) = -2 + \frac{4}{3}$$

To add the fraction and the integer, we need to find a common denominator. The least common multiple of 1 and 3 is 3, so we can rewrite the integer as a fraction with a denominator of 3:

$$f(-1) = -2 + \frac{4}{3} = -\frac{6}{3} + \frac{4}{3}$$

Now we can add the fractions:

$$f(-1) = -\frac{6}{3} + \frac{4}{3} = -\frac{2}{3}$$

Therefore, the value of $f(-1)$ is $-\frac{2}{3}$.

Conclusion

In this article, we found the value of $f(-1)$ for the given piecewise-defined function. We determined which sub-function to use by checking the interval of the domain, and then we substituted $x = -1$ into the first sub-function to find the value of $f(-1)$. The final answer is $-\frac{2}{3}$.

Additional Examples

To further illustrate the concept of piecewise-defined functions, let's consider a few more examples.

Example 1

Find the value of $f(2)$ for the piecewise-defined function:

$$f(x) = \begin{cases} x^2 + 1 & \text{if } x \ \textless \ 2 \\ 3x - 2 & \text{if } x \geq 2 \end{cases}$$

Since $2 \geq 2$, we will use the second sub-function: $3x - 2$. Substituting $x = 2$ into the second sub-function, we get:

$$f(2) = 3(2) - 2 = 6 - 2 = 4$$

Therefore, the value of $f(2)$ is 4.

Example 2

Find the value of $f(-4)$ for the piecewise-defined function:

$$f(x) = \begin{cases} 2x + 1 & \text{if } x \ \textless \ -2 \\ x^2 - 3 & \text{if } x \geq -2 \end{cases}$$

Since $-4 < -2$, we will use the first sub-function: $2x + 1$. Substituting $x = -4$ into the first sub-function, we get:

$$f(-4) = 2(-4) + 1 = -8 + 1 = -7$$

Therefore, the value of $f(-4)$ is -7.

Example 3

Find the value of $f(0)$ for the piecewise-defined function:

$$f(x) = \begin{cases} x^2 - 2 & \text{if } x \ \textless \ 0 \\ 2x + 3 & \text{if } x \geq 0 \end{cases}$$

Since $0 \geq 0$, we will use the second sub-function: $2x + 3$. Substituting $x = 0$ into the second sub-function, we get:

$$f(0) = 2(0) + 3 = 0 + 3 = 3$$

Therefore, the value of $f(0)$ is 3.

Conclusion

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Q: What is a piecewise-defined function?

A: A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In other words, a piecewise-defined function is a function that has different formulas for different parts of its domain.

Q: How do I determine which sub-function to use for a piecewise-defined function?

A: To determine which sub-function to use, you need to check the interval of the domain. If the input value is in the interval for the first sub-function, use that sub-function. If the input value is in the interval for the second sub-function, use that sub-function.

Q: Can I use a piecewise-defined function to model real-world situations?

A: Yes, piecewise-defined functions can be used to model real-world situations where the behavior of a system changes at certain points. For example, a piecewise-defined function can be used to model the cost of shipping a package, where the cost changes at certain weight thresholds.

Q: How do I find the value of a piecewise-defined function at a specific point?

A: To find the value of a piecewise-defined function at a specific point, you need to determine which sub-function to use and then substitute the input value into that sub-function.

Q: Can I graph a piecewise-defined function?

A: Yes, you can graph a piecewise-defined function by graphing each sub-function separately and then combining the graphs.

Q: How do I find the derivative of a piecewise-defined function?

A: To find the derivative of a piecewise-defined function, you need to find the derivative of each sub-function separately and then combine the derivatives.

Q: Can I use a piecewise-defined function to solve optimization problems?

A: Yes, piecewise-defined functions can be used to solve optimization problems where the objective function changes at certain points.

Q: How do I find the integral of a piecewise-defined function?

A: To find the integral of a piecewise-defined function, you need to find the integral of each sub-function separately and then combine the integrals.

Q: Can I use a piecewise-defined function to model population growth?

A: Yes, piecewise-defined functions can be used to model population growth where the growth rate changes at certain points.

Q: How do I find the maximum or minimum value of a piecewise-defined function?

A: To find the maximum or minimum value of a piecewise-defined function, you need to find the critical points of each sub-function separately and then compare the values.

Q: Can I use a piecewise-defined function to model economic systems?

A: Yes, piecewise-defined functions can be used to model economic systems where the behavior of the system changes at certain points.

Conclusion

In this article, we answered frequently asked questions about piecewise-defined functions. We covered topics such as determining which sub-function to use, finding the value of a piecewise-defined function at a specific point, graphing a piecewise-defined function, and using piecewise-defined functions to solve optimization problems. We also provided examples to illustrate the concepts.

Additional Resources

For more information on piecewise-defined functions, you can refer to the following resources:

• Wikipedia article on piecewise-defined functions
• MathWorld article on piecewise-defined functions
• Khan Academy video on piecewise-defined functions

Conclusion

In conclusion, piecewise-defined functions are a powerful tool for modeling real-world situations and solving mathematical problems. By understanding how to use piecewise-defined functions, you can gain a deeper understanding of mathematical concepts and develop problem-solving skills.