For The Following Exercise, Given The Function $f$, Evaluate $f(-3), F(-2), F(-1$\], And $f(0$\].$\[ \begin{array}{l} f(x) = \left\{ \begin{array}{ll} -2x^2 + 1 & \text{for } X \ \textless \ -1 \\ 6x - 5 & \text{for } X

Introduction

In this exercise, we are given a function $f(x)$ defined by a piecewise function. The function $f(x)$ is defined differently for different intervals of $x$. We are asked to evaluate the function $f(x)$ at four specific values: $-3$, $-2$, $-1$, and $0$. To evaluate the function at these values, we need to determine which interval each value falls into and then use the corresponding definition of the function.

The Piecewise Function

The function $f(x)$ is defined as follows:

$$f(x) = \left\{ \begin{array}{ll} -2x^2 + 1 & \text{for } x \ \textless \ -1 \\ 6x - 5 & \text{for } x \geq -1 \end{array} \right.$$

Evaluating f(-3)

To evaluate $f(-3)$, we need to determine which interval $-3$ falls into. Since $-3$ is less than $-1$, we use the first definition of the function:

$$f(-3) = -2(-3)^2 + 1$$

Using the order of operations, we first evaluate the exponent:

$$(-3)^2 = 9$$

Then, we multiply $-2$ by $9$:

$$-2(9) = -18$$

Finally, we add $1$ to get:

$$f(-3) = -18 + 1 = -17$$

Evaluating f(-2)

To evaluate $f(-2)$, we need to determine which interval $-2$ falls into. Since $-2$ is less than $-1$, we use the first definition of the function:

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

Using the order of operations, we first evaluate the exponent:

$$(-2)^2 = 4$$

Then, we multiply $-2$ by $4$:

$$-2(4) = -8$$

Finally, we add $1$ to get:

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

Evaluating f(-1)

To evaluate $f(-1)$, we need to determine which interval $-1$ falls into. Since $-1$ is equal to $-1$, we use the second definition of the function:

$$f(-1) = 6(-1) - 5$$

Using the order of operations, we first multiply $6$ by $-1$:

$$6(-1) = -6$$

Then, we subtract $5$ to get:

$$f(-1) = -6 - 5 = -11$$

Evaluating f(0)

To evaluate $f(0)$, we need to determine which interval $0$ falls into. Since $0$ is greater than or equal to $-1$, we use the second definition of the function:

$$f(0) = 6(0) - 5$$

Using the order of operations, we first multiply $6$ by $0$:

$$6(0) = 0$$

Then, we subtract $5$ to get:

$$f(0) = 0 - 5 = -5$$

Conclusion

In this exercise, we evaluated the function $f(x)$ at four specific values: $-3$, $-2$, $-1$, and $0$. We used the piecewise definition of the function to determine which interval each value falls into and then used the corresponding definition of the function to evaluate the function at each value. The results are as follows:

• $f(-3) = -17$
• $f(-2) = -7$
• $f(-1) = -11$
• $f(0) = -5$

Introduction

In our previous article, we evaluated the function $f(x)$ at four specific values: $-3$, $-2$, $-1$, and $0$. We used the piecewise definition of the function to determine which interval each value falls into and then used the corresponding definition of the function to evaluate the function at each value. In this article, we will answer some common questions related to the function $f(x)$ and its evaluation.

Q: What is the piecewise function?

A: The piecewise function is a function that is defined differently for different intervals of the input variable. In the case of the function $f(x)$, it is defined as follows:

$$f(x) = \left\{ \begin{array}{ll} -2x^2 + 1 & \text{for } x \ \textless \ -1 \\ 6x - 5 & \text{for } x \geq -1 \end{array} \right.$$

Q: How do I determine which interval a value falls into?

A: To determine which interval a value falls into, you need to compare the value to the boundary of the interval. In the case of the function $f(x)$, the boundary is $-1$. If the value is less than $-1$, it falls into the first interval, and if it is greater than or equal to $-1$, it falls into the second interval.

Q: What if the value is equal to the boundary?

A: If the value is equal to the boundary, it falls into the second interval. This is because the second interval includes all values greater than or equal to $-1$, including $-1$ itself.

Q: Can I use the same definition of the function for all values?

A: No, you cannot use the same definition of the function for all values. The piecewise function is defined differently for different intervals, and you need to use the corresponding definition of the function for each interval.

Q: How do I evaluate the function at a value that falls into the first interval?

A: To evaluate the function at a value that falls into the first interval, you need to use the first definition of the function:

$$f(x) = -2x^2 + 1$$

You can then plug in the value of $x$ into this equation to get the value of the function.

Q: How do I evaluate the function at a value that falls into the second interval?

A: To evaluate the function at a value that falls into the second interval, you need to use the second definition of the function:

$$f(x) = 6x - 5$$

You can then plug in the value of $x$ into this equation to get the value of the function.

Q: What if I make a mistake in evaluating the function?

A: If you make a mistake in evaluating the function, you may get an incorrect answer. To avoid this, make sure to follow the correct procedure for evaluating the function, and double-check your work.

Conclusion

In this article, we answered some common questions related to the function $f(x)$ and its evaluation. We discussed the piecewise function, how to determine which interval a value falls into, and how to evaluate the function at a value that falls into each interval. We also discussed what to do if you make a mistake in evaluating the function. By following these guidelines, you can ensure that you evaluate the function correctly and get the correct answer.