La cantidad de millones de bacterias vivas en un cultivo, después de aplicar un tratamiento, está dada por la expresión
[tex]$f(x)=1+\neq \overline{10 \pm x}$[/tex]
Donde [tex]$x$[/tex] es el número de días que han pasado desde que se aplicó el tratamiento. ¿Cuál es el conjunto de todos los valores de [tex]$x$[/tex] para el cual [tex]$\ddot{i}(x)$[/tex] está definida?
A. [tex]$x \cdot 10$[/tex]
B. [tex]$x\ \textgreater \ 10$[/tex]
C. [tex]$0^{\prime \prime} x$[/tex] " 10
D. [tex]$0\ \textless \ x\ \textless \ 10$[/tex]
Answer (2)
The function is defined when the expression inside the square root is non-negative: 10 − x ≥ 0 .
Solve the inequality: x ≤ 10 .
Since x represents the number of days, it must be non-negative: x ≥ 0 .
Combine both conditions: 0 ≤ x ≤ 10 , so the answer is 0 ≤ x ≤ 10 .
Explanation
Understanding the Problem The function given is f ( x ) = 1 + 10 − x . We need to find the domain of this function, which means we need to find all possible values of x for which the function is defined.
Setting up the Inequality The square root function is only defined for non-negative values. Therefore, the expression inside the square root must be greater than or equal to zero: 10 − x ≥ 0
Solving the Inequality Now, we solve the inequality for x :
10 − x ≥ 0 10 ≥ x x ≤ 10
Considering the Non-Negativity of Days Since x represents the number of days, it must be non-negative. Therefore, we also have the condition: x ≥ 0
Combining the Conditions Combining both conditions, we have: 0 ≤ x ≤ 10
Final Answer The set of all values of x for which f ( x ) is defined is 0 ≤ x ≤ 10 . This corresponds to option C: 0 ≤ x ≤ 10 .
Examples
Understanding the domain of a function like f ( x ) = 1 + 10 − x is crucial in many real-world scenarios. For example, if x represents the number of days after applying a treatment to a bacterial culture, and f ( x ) represents the number of millions of live bacteria, then knowing the domain tells us for how many days the model is valid. If x goes beyond 10, the model no longer makes sense because we can't have a negative number of bacteria due to the square root. This is also applicable in finance, where you might model the growth of an investment over time, and the domain tells you the period for which the model is reliable.
The function f ( x ) = 1 + 10 − x is defined for values of x within the range of 0 to 10 days, resulting in the constraint 0 ≤ x ≤ 10 . Therefore, the chosen option is D: 0 < x < 10 although 10 is included as the upper limit. This indicates that the model is valid up to 10 days after treatment.
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