For The Following Exercise, Given The Function F F F , Evaluate F ( − 1 F(-1 F ( − 1 ], F ( 0 F(0 F ( 0 ], F ( 2 F(2 F ( 2 ], And F ( 4 F(4 F ( 4 ].$[ f(x) = \begin{cases} 2x & \text{for } X \ \textless \ 0 \ 3 & \text{for } 0 \leq X \leq 3 \ x^2

In this exercise, we are given a function $f(x)$ and asked to evaluate it at specific values of $x$. The function $f(x)$ is defined as:

$$f(x) = \begin{cases} 2x & \text{for } x < 0 \\ 3 & \text{for } 0 \leq x \leq 3 \\ x^2 & \text{for } x > 3 \end{cases}$$

To evaluate the function at specific values of $x$, we need to determine which part of the piecewise function applies to each value of $x$. We will then substitute the value of $x$ into the corresponding part of the function to find the value of $f(x)$.

Evaluating f(-1)

To evaluate $f(-1)$, we need to determine which part of the piecewise function applies to $x = -1$. Since $x = -1$ is less than 0, we use the first part of the function, which is $f(x) = 2x$. Substituting $x = -1$ into this part of the function, we get:

$$f(-1) = 2(-1) = -2$$

Therefore, $f(-1) = -2$.

Evaluating f(0)

To evaluate $f(0)$, we need to determine which part of the piecewise function applies to $x = 0$. Since $x = 0$ is between 0 and 3 (inclusive), we use the second part of the function, which is $f(x) = 3$. Substituting $x = 0$ into this part of the function, we get:

$$f(0) = 3$$

Therefore, $f(0) = 3$.

Evaluating f(2)

To evaluate $f(2)$, we need to determine which part of the piecewise function applies to $x = 2$. Since $x = 2$ is between 0 and 3 (inclusive), we use the second part of the function, which is $f(x) = 3$. Substituting $x = 2$ into this part of the function, we get:

$$f(2) = 3$$

Therefore, $f(2) = 3$.

Evaluating f(4)

To evaluate $f(4)$, we need to determine which part of the piecewise function applies to $x = 4$. Since $x = 4$ is greater than 3, we use the third part of the function, which is $f(x) = x^2$. Substituting $x = 4$ into this part of the function, we get:

$$f(4) = 4^2 = 16$$

Therefore, $f(4) = 16$.

Conclusion

In this exercise, we evaluated the function $f(x)$ at specific values of $x$. We determined which part of the piecewise function applied to each value of $x$ and substituted the value of $x$ into the corresponding part of the function to find the value of $f(x)$. The values of $f(x)$ at $x = -1$, $x = 0$, $x = 2$, and $x = 4$ are $-2$, $3$, $3$, and $16$, respectively.

Key Takeaways

• To evaluate a piecewise function at a specific value of $x$, we need to determine which part of the function applies to that value of $x$.
• We can substitute the value of $x$ into the corresponding part of the function to find the value of the function at that value of $x$.
• The function $f(x)$ is defined as a piecewise function, which means it has different definitions for different intervals of $x$.

Further Exploration

• What happens to the function $f(x)$ as $x$ approaches 0 from the left and from the right?
• How does the function $f(x)$ behave as $x$ increases without bound?
• Can we find a pattern in the values of $f(x)$ as $x$ increases without bound?

References

• [1] "Piecewise Functions." Math Open Reference, mathopenref.com/piecewise.html.
• [2] "Evaluating Piecewise Functions." Purplemath, purplemath.com/modules/piecewise.htm.

$$

In this article, we evaluated the function $f(x)$ at specific values of $x$. We determined which part of the piecewise function applied to each value of $x$ and substituted the value of $x$ into the corresponding part of the function to find the value of $f(x)$. In this Q&A section, we will answer some common questions related to the function $f(x)$.

Q: What is a piecewise function?

A: A piecewise function is a function that is defined by multiple sub-functions, each of which is applied to a specific interval of the input variable. In the case of the function $f(x)$, it is defined as:

$$f(x) = \begin{cases} 2x & \text{for } x < 0 \\ 3 & \text{for } 0 \leq x \leq 3 \\ x^2 & \text{for } x > 3 \end{cases}$$

Q: How do I determine which part of the piecewise function applies to a given value of x?

A: To determine which part of the piecewise function applies to a given value of $x$, you need to compare the value of $x$ to the intervals defined in the function. For example, if $x = 2$, you would compare $x$ to the intervals $x < 0$, $0 \leq x \leq 3$, and $x > 3$. Since $x = 2$ is between 0 and 3 (inclusive), you would use the second part of the function, which is $f(x) = 3$.

Q: What happens if x is equal to one of the endpoints of an interval?

A: If $x$ is equal to one of the endpoints of an interval, you need to use the corresponding part of the function. For example, if $x = 0$, you would use the second part of the function, which is $f(x) = 3$. Similarly, if $x = 3$, you would also use the second part of the function, which is $f(x) = 3$.

Q: Can I use the same part of the function for multiple intervals?

A: No, you cannot use the same part of the function for multiple intervals. Each part of the function is defined for a specific interval, and you need to use the corresponding part of the function for each interval.

Q: How do I evaluate the function f(x) at a value of x that is not in any of the intervals?

A: If $x$ is not in any of the intervals, you need to check if it is less than 0, between 0 and 3 (inclusive), or greater than 3. If it is less than 0, you would use the first part of the function, which is $f(x) = 2x$. If it is between 0 and 3 (inclusive), you would use the second part of the function, which is $f(x) = 3$. If it is greater than 3, you would use the third part of the function, which is $f(x) = x^2$.

Q: Can I simplify the function f(x) by combining the different parts?

A: Yes, you can simplify the function $f(x)$ by combining the different parts. For example, you can rewrite the function as:

$$f(x) = \begin{cases} 2x & \text{for } x < 0 \\ 3 & \text{for } 0 \leq x \leq 3 \\ x^2 & \text{for } x > 3 \end{cases} = \begin{cases} 2x & \text{for } x < 0 \\ 3 & \text{for } 0 \leq x \leq 3 \\ x^2 & \text{for } x > 3 \end{cases}$$

However, this simplification does not change the fact that the function is defined by multiple sub-functions, each of which is applied to a specific interval of the input variable.

Q: What are some common applications of piecewise functions?

A: Piecewise functions have many common applications in mathematics, science, and engineering. Some examples include:

• Modeling real-world phenomena that have different behaviors in different intervals, such as the cost of a product that changes depending on the quantity ordered.
• Representing data that has different patterns in different intervals, such as the population growth of a species that changes depending on the availability of resources.
• Solving problems that involve different equations in different intervals, such as the motion of an object that changes depending on the force applied.

Conclusion

In this Q&A section, we answered some common questions related to the function $f(x)$. We discussed how to determine which part of the piecewise function applies to a given value of $x$, how to evaluate the function at a value of $x$ that is not in any of the intervals, and how to simplify the function by combining the different parts. We also discussed some common applications of piecewise functions.

Key Takeaways

• To evaluate a piecewise function at a specific value of $x$, you need to determine which part of the function applies to that value of $x$.
• You can use the corresponding part of the function for each interval.
• Piecewise functions have many common applications in mathematics, science, and engineering.

Further Exploration

• What are some other examples of piecewise functions?
• How can you use piecewise functions to model real-world phenomena?
• What are some common challenges when working with piecewise functions?