Question
Select the correct answer.
Which statement describes the end behavior of the function f(x) = 3|x − 7| − 7?
A.
As x approaches negative infinity, f(x) approaches negative infinity.
B.
As x approaches negative infinity, f(x) approaches positive infinity.
C.
As x approaches positive infinity, f(x) approaches negative infinity.
D.
As x approaches positive infinity, f(x) is no longer continuous.
Which statement describes the end behavior of the function f(x) = 3|x − 7| − 7?
A.
As x approaches negative infinity, f(x) approaches negative infinity.
B.
As x approaches negative infinity, f(x) approaches positive infinity.
C.
As x approaches positive infinity, f(x) approaches negative infinity.
D.
As x approaches positive infinity, f(x) is no longer continuous.
Answers
GPT-4o mini
To determine the end behavior of the function \( f(x) = 3|x - 7| - 7 \), we need to evaluate what happens to \( f(x) \) as \( x \) approaches both negative infinity and positive infinity.
1. **As \( x \) approaches negative infinity**:
- The expression \( |x - 7| \) behaves like \( |x| \) because \( x - 7 \) will be negative as \( x \) becomes very negative. Thus, \( |x - 7| = -(x - 7) = -x + 7 \).
- Therefore, \( f(x) = 3(-x + 7) - 7 = -3x + 21 - 7 = -3x + 14 \).
- As \( x \) goes to negative infinity (\( x \to -\infty \)), \( -3x \) goes to positive infinity. Hence, \( f(x) \) approaches positive infinity.
2. **As \( x \) approaches positive infinity**:
- When \( x \) is very large (positive), \( |x - 7| \) becomes simply \( x - 7 \) since \( x - 7 \) is positive for large \( x \).
- Therefore, \( f(x) = 3(x - 7) - 7 = 3x - 21 - 7 = 3x - 28 \).
- As \( x \) approaches positive infinity (\( x \to \infty \)), \( 3x - 28 \) also approaches positive infinity.
Given this analysis, we summarize the end behavior as follows:
- As \( x \) approaches negative infinity, \( f(x) \) approaches positive infinity.
- As \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity.
Thus, the correct statement from the options provided is:
**B. As x approaches negative infinity, f(x) approaches positive infinity.**
1. **As \( x \) approaches negative infinity**:
- The expression \( |x - 7| \) behaves like \( |x| \) because \( x - 7 \) will be negative as \( x \) becomes very negative. Thus, \( |x - 7| = -(x - 7) = -x + 7 \).
- Therefore, \( f(x) = 3(-x + 7) - 7 = -3x + 21 - 7 = -3x + 14 \).
- As \( x \) goes to negative infinity (\( x \to -\infty \)), \( -3x \) goes to positive infinity. Hence, \( f(x) \) approaches positive infinity.
2. **As \( x \) approaches positive infinity**:
- When \( x \) is very large (positive), \( |x - 7| \) becomes simply \( x - 7 \) since \( x - 7 \) is positive for large \( x \).
- Therefore, \( f(x) = 3(x - 7) - 7 = 3x - 21 - 7 = 3x - 28 \).
- As \( x \) approaches positive infinity (\( x \to \infty \)), \( 3x - 28 \) also approaches positive infinity.
Given this analysis, we summarize the end behavior as follows:
- As \( x \) approaches negative infinity, \( f(x) \) approaches positive infinity.
- As \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity.
Thus, the correct statement from the options provided is:
**B. As x approaches negative infinity, f(x) approaches positive infinity.**