Answered

Given that a function, $g$, has a domain of $-20 \leq x \leq 5$ and a range of $-5 \leq g(x) \leq 45$, and that $g(0)=-2$ and $g(-9)=6$, select the statement that could be true for $g$.


A$g(7)=-1$
B$g(-13)=20$
C$g(-4)=-11$
D$g(0)=2$

Verified Answer :

  • The domain \(-20 \leq x \leq 5\) excludes \(g(7) = -1\) because \(7 > 5\).
  • The range \(-5 \leq g(x) \leq 45\) excludes \(g(-4) = -11\) because \(-11 < -5\).
  • The function definition excludes \(g(0) = 2\) because \(g(0) = -2\) is given.
  • The statement \(g(-13) = 20\) satisfies both the domain and the range requirements.
  • Final Answer: \(\boxed{g(-13) = 20}\)

Explanation

  1. Problem Analysis and Data Overview

We are analyzing the properties of a function \(g\) with a given domain and range, along with two specific coordinate points. The domain of the function is the set of all possible input values \(x\), which is \(-20 \leq x \leq 5\). The range of the function is the set of all possible output values \(g(x)\), which is \(-5 \leq g(x) \leq 45\). Additionally, we are given two defined points on the function: \(g(0) = -2\) and \(g(-9) = 6\). Our task is to evaluate four given statements and determine which one could potentially be true for this function.

  1. Evaluating the Statements

We evaluate each statement by checking if the input \(x\) lies within the domain and if the output \(g(x)\) lies within the specified range, while also respecting the function property (one input maps to exactly one output).

  1. Checking \(g(7) = -1\): The input is \(x = 7\). Since the domain of \(g\) is \(-20 \leq x \leq 5\), the value \(x = 7\) is outside the domain. Thus, \(g(7)\) cannot exist.
  2. Checking \(g(-13) = 20\): The input is \(x = -13\). Since \(-20 \leq -13 \leq 5\), this is within the domain. The output is \(g(-13) = 20\). Since \(-5 \leq 20 \leq 45\), this is within the range. This statement does not violate any given conditions.
  3. Checking \(g(-4) = -11\): The input is \(x = -4\). This is within the domain. The output is \(g(-4) = -11\). However, the range of \(g\) is \(-5 \leq g(x) \leq 45\). Since \(-11 < -5\), this output is outside the possible range.
  4. Checking \(g(0) = 2\): We are given that \(g(0) = -2\). By the definition of a function, each input must correspond to exactly one output. Since \(g(0)\) is already defined as \(-2\), it cannot simultaneously be \(2\).
  1. Conclusion

Based on our evaluation:

  • \(g(7) = -1\) is invalid because \(7\) is not in the domain.
  • \(g(-13) = 20\) is valid as both \(-13\) is in the domain and \(20\) is in the range.
  • \(g(-4) = -11\) is invalid because \(-11\) is outside the range.
  • \(g(0) = 2\) is invalid because \(g(0)\) is already defined as \(-2\).

Therefore, the only statement that could be true for the function \(g\) is \(g(-13) = 20\). The final answer is \(\boxed{g(-13) = 20}\).

Examples

Consider a thermostat system in a building where the temperature, denoted by \(g(x)\), depends on the time of day, \(x\). The domain represents the operating hours, e.g., \(-20\) to \(5\) (in relative time units), and the range represents the allowed temperature settings, e.g., \(-5^\circ C\) to \(45^\circ C\). If the system is programmed with certain fixed settings at specific times (like the given points \(g(0)=-2\) and \(g(-9)=6\)), checking whether a proposed temperature at another time is "valid" helps in verifying the system's logic and ensuring it stays within its operational safety limits.

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