To determine the end behavior of the function [tex]\( f(x) = 3|x - 7| - 7 \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches negative infinity and positive infinity.
1. As [tex]\( x \)[/tex] approaches negative infinity:
- When [tex]\( x \)[/tex] becomes very large in the negative direction, [tex]\( x - 7 \)[/tex] is also very large in the negative direction.
- The absolute value [tex]\( |x - 7| \)[/tex] converts this large negative value into a large positive value. Therefore, [tex]\( |x - 7| \)[/tex] approaches positive infinity as [tex]\( x \)[/tex] approaches negative infinity.
- Multiplying this by 3 yields a very large positive value, so [tex]\( 3|x - 7| \)[/tex] also approaches positive infinity.
- Subtracting 7 from a very large positive value still results in a large positive value.
Therefore, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
So, the correct answer corresponding to this behavior is:
[tex]\[
B. \text{As } x \text{ approaches negative infinity, } f(x) \text{ approaches positive infinity.}
\][/tex]
2. As [tex]\( x \)[/tex] approaches positive infinity:
- When [tex]\( x \)[/tex] becomes very large in the positive direction, [tex]\( x - 7 \)[/tex] is also very large.
- The absolute value [tex]\( |x - 7| \)[/tex] remains a large positive value as [tex]\( x \)[/tex] approaches positive infinity.
- Multiplying this by 3 yields a very large positive value, so [tex]\( 3|x - 7| \)[/tex] also approaches positive infinity.
- Subtracting 7 from a very large positive value still results in a large positive value.
Therefore, as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
So, the correct answer corresponding to this behavior is:
[tex]\[
B. \text{As } x \text{ approaches positive infinity, } f(x) \text{ approaches positive infinity.}
\][/tex]
Hence, putting everything together, the statements that correctly describe the end behavior of the function [tex]\( f(x) = 3|x - 7| - 7 \)[/tex] are:
1. [tex]\( B \)[/tex] As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
2. [tex]\( B \)[/tex] As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
