Sure, let's analyze the function [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex] and understand its behavior step by step.
1. Understanding the Function [tex]\( \sqrt{x^2} \)[/tex]:
- The term [tex]\( \sqrt{x^2} \)[/tex] represents the absolute value of [tex]\( x \)[/tex].
- For any real number [tex]\( x \)[/tex], [tex]\( \sqrt{x^2} \)[/tex] is equivalent to [tex]\( |x| \)[/tex], since the square root of a square always returns the non-negative value.
- Therefore, we can rewrite the function [tex]\( f(x) \)[/tex] as [tex]\( f(x) = |x| - 9 \)[/tex].
2. Evaluating [tex]\( f(x) \)[/tex]:
- To further understand the function [tex]\( f(x) = |x| - 9 \)[/tex], we should consider the behavior of the absolute value function.
- The absolute value function [tex]\( |x| \)[/tex] is defined as:
[tex]\[
|x| =
\begin{cases}
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{cases}
\][/tex]
3. Analyzing [tex]\( f(x) = |x| - 9 \)[/tex]:
- For [tex]\( x \geq 0 \)[/tex]:
[tex]\[
f(x) = x - 9
\][/tex]
- For [tex]\( x < 0 \)[/tex]:
[tex]\[
f(x) = -x - 9
\][/tex]
This gives us a piecewise function:
[tex]\[
f(x) =
\begin{cases}
x - 9 & \text{if } x \geq 0 \\
-x - 9 & \text{if } x < 0
\end{cases}
\][/tex]
4. Evaluating [tex]\( f(x) \)[/tex] at Specific Points:
- Let's evaluate [tex]\( f(x) \)[/tex] for some specific values of [tex]\( x \)[/tex]:
- If [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 3 - 9 = -6
\][/tex]
- If [tex]\( x = -3 \)[/tex]:
[tex]\[
f(-3) = -(-3) - 9 = 3 - 9 = -6
\][/tex]
- If [tex]\( x = 9 \)[/tex]:
[tex]\[
f(9) = 9 - 9 = 0
\][/tex]
- If [tex]\( x = -9 \)[/tex]:
[tex]\[
f(-9) = -(-9) - 9 = 9 - 9 = 0
\][/tex]
5. General Behavior:
- For [tex]\( x \geq 9 \)[/tex], [tex]\( f(x) = x - 9 \)[/tex] is positive or zero.
- For [tex]\( x \leq -9 \)[/tex], [tex]\( f(x) = -x - 9 \)[/tex] is positive or zero.
- For [tex]\(|x| < 9\)[/tex], [tex]\( f(x) = |x| - 9 \)[/tex] is negative.
Hence, the function [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex] can be described as above, encompassing both positive and negative values of [tex]\( x \)[/tex] with its respective evaluations.
