B) $f(x) = X^2 - 2$Complete The Table By Calculating $f(x$\] For Each Value Of $x$:$\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & & & & & & &

$$

Introduction

In mathematics, functions are used to describe the relationship between variables. A function is a rule that assigns to each input value, or independent variable, exactly one output value, or dependent variable. In this article, we will focus on the function $f(x) = x^2 - 2$ and complete a table by calculating $f(x)$ for each value of $x$.

The Function $f(x) = x^2 - 2$

The given function is a quadratic function, which is a polynomial function of degree two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, the function is $f(x) = x^2 - 2$, where $a = 1$, $b = 0$, and $c = -2$.

Completing the Table

To complete the table, we need to calculate $f(x)$ for each value of $x$. We will substitute each value of $x$ into the function and simplify the expression to find the corresponding value of $f(x)$.

Calculating $f(x)$ for $x = -3$

$f(-3) = (-3)^2 - 2$ $f(-3) = 9 - 2$ $f(-3) = 7$

Calculating $f(x)$ for $x = -2$

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

Calculating $f(x)$ for $x = -1$

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

Calculating $f(x)$ for $x = 0$

$f(0) = (0)^2 - 2$ $f(0) = 0 - 2$ $f(0) = -2$

Calculating $f(x)$ for $x = 1$

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

Calculating $f(x)$ for $x = 2$

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

Calculating $f(x)$ for $x = 3$

$f(3) = (3)^2 - 2$ $f(3) = 9 - 2$ $f(3) = 7$

Completed Table

$x$	$f(x)$
-3	7
-2	2
-1	-1
0	-2
1	-1
2	2
3	7

Discussion

The completed table shows the values of $f(x)$ for each value of $x$. We can see that the function $f(x) = x^2 - 2$ is a quadratic function that has a minimum value of $-2$ at $x = 0$. The function is symmetric about the line $x = 0$, and the values of $f(x)$ are positive for $x > 0$ and negative for $x < 0$.

Conclusion

In this article, we completed the table for the function $f(x) = x^2 - 2$ by calculating $f(x)$ for each value of $x$. We also discussed the properties of the function, including its minimum value and symmetry about the line $x = 0$. The completed table provides a visual representation of the function and can be used to help understand its behavior.

References

• [1] "Functions" by Khan Academy
• [2] "Quadratic Functions" by Math Open Reference

Additional Resources

• [1] "Functions" by Wolfram MathWorld
• [2] "Quadratic Functions" by Purplemath
  Q&A: Understanding the Function $f(x) = x^2 - 2$ =====================================================

Introduction

In our previous article, we completed the table for the function $f(x) = x^2 - 2$ and discussed its properties. In this article, we will answer some frequently asked questions about the function and provide additional insights into its behavior.

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

A: The domain of a function is the set of all possible input values for which the function is defined. In this case, the function $f(x) = x^2 - 2$ is defined for all real numbers, so the domain is all real numbers, denoted as $(-\infty, \infty)$.

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

A: The range of a function is the set of all possible output values for which the function is defined. In this case, the function $f(x) = x^2 - 2$ is a quadratic function that has a minimum value of $-2$ at $x = 0$. Therefore, the range is all real numbers greater than or equal to $-2$, denoted as $[-2, \infty)$.

Q: Is the function $f(x) = x^2 - 2$ an even function?

A: An even function is a function that satisfies the condition $f(-x) = f(x)$ for all $x$ in the domain. In this case, we can substitute $-x$ into the function and simplify:

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

Therefore, the function $f(x) = x^2 - 2$ is an even function.

Q: Is the function $f(x) = x^2 - 2$ a one-to-one function?

A: A one-to-one function is a function that satisfies the condition $f(x_1) = f(x_2)$ implies $x_1 = x_2$ for all $x_1$ and $x_2$ in the domain. In this case, we can see that the function $f(x) = x^2 - 2$ is not one-to-one because it has a minimum value of $-2$ at $x = 0$, and there are multiple values of $x$ that map to the same value of $f(x)$.

Q: How can we graph the function $f(x) = x^2 - 2$?

A: To graph the function $f(x) = x^2 - 2$, we can use the following steps:

1. Plot the point $(0, -2)$, which is the minimum value of the function.
2. Plot the points $(x, f(x))$ for $x = -3, -2, -1, 1, 2, 3$, which are the values of $x$ that we calculated in our previous article.
3. Draw a smooth curve through the points to obtain the graph of the function.

Conclusion

In this article, we answered some frequently asked questions about the function $f(x) = x^2 - 2$ and provided additional insights into its behavior. We also discussed how to graph the function and its properties, including its domain, range, and symmetry.

References

• [1] "Functions" by Khan Academy
• [2] "Quadratic Functions" by Math Open Reference

Additional Resources

• [1] "Functions" by Wolfram MathWorld
• [2] "Quadratic Functions" by Purplemath