Select the correct answer.

Consider functions [tex]$f$[/tex] and [tex]$g$[/tex].
[tex]
\begin{array}{l}
f(x)=\log ^{\prime}(x+2) \\
g(x)=4^x-1
\end{array}
[/tex]
Using a table of values, what is the approximate positive solution to the equation [tex]$f(x)=g(x)$[/tex] to the nearest quarter of a unit?
A. [tex]$x \approx 1.75$[/tex]
B. [tex]$z \approx 0.50$[/tex]
C. [tex]$x \approx 0.75$[/tex]
D. [tex]$x \approx 0.25$[/tex]

Create a table of values for f ( x ) = lo g ( x + 2 ) and g ( x ) = 4 x − 1 for x = 0 , 0.25 , 0.50 , 0.75 , 1.00 , 1.25 , 1.50 , 1.75 , 2.00 .
Calculate the absolute difference ∣ f ( x ) − g ( x ) ∣ for each value of x .
Identify the value of x where the absolute difference is the smallest.
The smallest absolute difference is 0.0620, which occurs at x = 0.25 , so the answer is 0.25 ​ .

Explanation

Understanding the Problem We are given two functions, f ( x ) = lo g ( x + 2 ) and g ( x ) = 4 x − 1 , and we want to find the approximate positive solution to the equation f ( x ) = g ( x ) to the nearest quarter of a unit using a table of values. This means we need to find the value of x for which f ( x ) and g ( x ) are closest to each other. The possible answers are A. x ≈ 1.75 , B. x ≈ 0.50 , C. x ≈ 0.75 , D. x ≈ 0.25 .

Creating a Table of Values We will create a table of values for f ( x ) and g ( x ) for x = 0 , 0.25 , 0.50 , 0.75 , 1.00 , 1.25 , 1.50 , 1.75 , 2.00 . Then, we will calculate f ( x ) = lo g ( x + 2 ) and g ( x ) = 4 x − 1 for each value of x . Finally, we will compare the values of f ( x ) and g ( x ) in the table and identify the value of x where f ( x ) and g ( x ) are closest to each other.

Calculating Function Values Using the python calculation tool, we have the following results:


x = 0.00, f(x) = 0.3010, g(x) = 0.0000, |f(x) - g(x)| = 0.3010 x = 0.25, f(x) = 0.3522, g(x) = 0.4142, |f(x) - g(x)| = 0.0620 x = 0.50, f(x) = 0.3979, g(x) = 1.0000, |f(x) - g(x)| = 0.6021 x = 0.75, f(x) = 0.4393, g(x) = 1.8284, |f(x) - g(x)| = 1.3891 x = 1.00, f(x) = 0.4771, g(x) = 3.0000, |f(x) - g(x)| = 2.5229 x = 1.25, f(x) = 0.5119, g(x) = 4.6569, |f(x) - g(x)| = 4.1450 x = 1.50, f(x) = 0.5441, g(x) = 7.0000, |f(x) - g(x)| = 6.4559 x = 1.75, f(x) = 0.5740, g(x) = 10.3137, |f(x) - g(x)| = 9.7397 x = 2.00, f(x) = 0.6021, g(x) = 15.0000, |f(x) - g(x)| = 14.3979
We are looking for the smallest absolute difference between f ( x ) and g ( x ) .

Finding the Closest Values From the table, we can see that the smallest absolute difference between f ( x ) and g ( x ) is 0.0620, which occurs at x = 0.25 . Therefore, the approximate positive solution to the equation f ( x ) = g ( x ) to the nearest quarter of a unit is x ≈ 0.25 .

Final Answer The approximate positive solution to the equation f ( x ) = g ( x ) to the nearest quarter of a unit is x ≈ 0.25 .


Examples
In drug dosage calculation, finding the intersection of logarithmic and exponential functions can help determine the optimal drug concentration in the bloodstream over time. The logarithmic function might represent drug absorption, while the exponential function represents drug decay. By finding the point where these functions are equal, medical professionals can determine the time at which the drug's concentration is most effective. This ensures that patients receive the maximum therapeutic benefit while minimizing potential side effects. This approach is crucial for drugs with narrow therapeutic windows, where precise dosage is essential for patient safety and treatment efficacy.

Answered by GinnyAnswer | 2025-07-08