Select the correct answer.
Given that a function, [tex]g[/tex], has a domain of [tex]$-20 \leq x \leq 5$[/tex] and a range of [tex]$-5 \leq g(x) \leq 45$[/tex] and that [tex]$g(0)=-2$[/tex] and [tex]$g(-9)=6$[/tex], select the statement that could be true for [tex]g[/tex].
A. [tex]$g(7)=-1$[/tex]
B. [tex]$g(-13)=20$[/tex]
C. [tex]$g(0)=2$[/tex]
D. [tex]$g(-4)=-11[/tex]
Answer (2)
Check if the x-value is within the domain − 20 \[ x ≤ 5 .
Check if the g(x) value is within the range − 5 \[ g ( x ) ≤ 45 .
Option A is false because 7 is not in the domain.
Option B could be true because − 13 is in the domain and 20 is in the range. The final answer is B .
Explanation
Understanding the Problem We are given a function g with a domain of − 20 \[ x ≤ 5 and a range of − 5 \[ g ( x ) ≤ 45 . We also know that g ( 0 ) = − 2 and g ( − 9 ) = 6 . We need to determine which of the given statements could be true.
Analyzing the Options Let's analyze each option:
Option A: g ( 7 ) = − 1 . Since the domain of g is − 20 \[ x ≤ 5 , x = 7 is not in the domain. Therefore, this statement cannot be true.
Option B: g ( − 13 ) = 20 . Since − 20 \[ − 13 ≤ 5 , − 13 is in the domain. Also, − 5 \[ 20 ≤ 45 , so 20 is in the range. This statement does not contradict the given information, so it could be true.
Option C: g ( 0 ) = 2 . We are given that g ( 0 ) = − 2 . This statement contradicts the given information, so it cannot be true.
Option D: g ( − 4 ) = − 11 . Since − 20 \[ − 4 ≤ 5 , − 4 is in the domain. However, − 5 \[ g ( x ) ≤ 45 , and − 11 is not in this range. Therefore, this statement cannot be true.
Conclusion Based on the analysis, only option B could be true.
Examples
Understanding function domains and ranges is crucial in many real-world applications. For example, consider a function that models the temperature of a chemical reaction over time. The domain represents the time interval during which the reaction is observed, and the range represents the possible temperature values. Knowing the domain and range helps scientists predict and control the reaction process, ensuring safety and efficiency. Similarly, in economics, functions can model the relationship between investment and profit, with the domain representing the possible investment amounts and the range representing the potential profit levels. By understanding these concepts, economists can make informed decisions about resource allocation and investment strategies.
The correct answer is option B: g ( − 13 ) = 20 , as it is the only statement that fits within the specified domain of g and its range. The other options either lie outside the domain or contradict the given values of the function. Therefore, only option B could potentially be true.
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