To determine the end behavior of the function [tex]\( f(x) = 3|x-7| - 7 \)[/tex], we need to consider what happens to the function as [tex]\( x \)[/tex] approaches positive infinity and negative infinity.
1. As [tex]\( x \)[/tex] approaches positive infinity:
When [tex]\( x \)[/tex] is very large and positive, the expression [tex]\( |x-7| \)[/tex] simplifies to [tex]\( x-7 \)[/tex] because the absolute value of a large positive number remains positive.
So, [tex]\( f(x) = 3|x-7| - 7 \)[/tex] becomes:
[tex]\[
f(x) = 3(x-7) - 7
\][/tex]
Simplify this expression:
[tex]\[
f(x) = 3x - 21 - 7
\][/tex]
[tex]\[
f(x) = 3x - 28
\][/tex]
As [tex]\( x \)[/tex] approaches positive infinity, the term [tex]\( 3x \)[/tex] will dominate, causing [tex]\( f(x) \)[/tex] to also approach positive infinity.
2. As [tex]\( x \)[/tex] approaches negative infinity:
When [tex]\( x \)[/tex] is very large and negative, the expression [tex]\( |x-7| \)[/tex] simplifies to [tex]\( -(x-7) \)[/tex] because the absolute value of a large negative number becomes positive when you take the negative of it.
So, [tex]\( f(x) = 3|x-7| - 7 \)[/tex] becomes:
[tex]\[
f(x) = 3(-(x-7)) - 7
\][/tex]
Simplify this expression:
[tex]\[
f(x) = 3(-x + 7) - 7
\][/tex]
[tex]\[
f(x) = -3x + 21 - 7
\][/tex]
[tex]\[
f(x) = -3x + 14
\][/tex]
As [tex]\( x \)[/tex] approaches negative infinity, the term [tex]\( -3x \)[/tex] will dominate, causing [tex]\( f(x) \)[/tex] to approach negative infinity.
Based on this analysis, we can conclude:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
Therefore, the correct answer is:
A. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
