Select the correct answer.
What is the range of the function $f(x)=-4|x+1|-5$?
A. $(-\infty,-5]$
B. $[-5, \infty)$
C. $[-4, \infty)$
D. $(-\infty,-4]$
Answer (1)
The function is f ( x ) = − 4∣ x + 1∣ − 5 .
The absolute value term ∣ x + 1∣ is always non-negative, so − 4∣ x + 1∣ is always non-positive.
Therefore, f ( x ) = − 4∣ x + 1∣ − 5 is always less than or equal to − 5 .
The range of f ( x ) is ( − ∞ , − 5 ] . ( − ∞ , − 5 ]
Explanation
Understanding the Function We are given the function f ( x ) = − 4∣ x + 1∣ − 5 and asked to find its range. The range of a function is the set of all possible output values (y-values).
Analyzing the Absolute Value The absolute value function ∣ x + 1∣ is always non-negative, meaning ∣ x + 1∣ ≥ 0 for all real numbers x .
Considering the Negative Coefficient Since ∣ x + 1∣ ≥ 0 , multiplying by − 4 reverses the inequality, so − 4∣ x + 1∣ ≤ 0 .
Determining the Upper Bound Subtracting 5 from both sides, we get − 4∣ x + 1∣ − 5 ≤ − 5 . This means that f ( x ) ≤ − 5 .
Finding the Maximum Value The maximum value of f ( x ) occurs when ∣ x + 1∣ = 0 , which happens when x = − 1 . In this case, f ( − 1 ) = − 4∣ ( − 1 ) + 1∣ − 5 = − 4 ( 0 ) − 5 = − 5 .
Determining the Range As x moves away from − 1 , ∣ x + 1∣ increases, and − 4∣ x + 1∣ becomes more negative. Thus, f ( x ) decreases without bound. This means that f ( x ) can take any value less than or equal to − 5 .
Final Answer Therefore, the range of f ( x ) is ( − ∞ , − 5 ] .
Examples
Understanding the range of functions like f ( x ) = − 4∣ x + 1∣ − 5 is crucial in various real-world scenarios. For instance, if f ( x ) represents the profit of a business, where x is the number of units sold, knowing the range helps determine the maximum possible loss the business can incur. In this case, the maximum loss is limited to -5 (in some currency unit), providing a clear boundary for financial planning and risk assessment. This concept extends to other areas like physics, where it can define the boundaries of a system's energy or displacement.
