Let's solve the given problem step-by-step.
1. Define the Function:
The function [tex]\( f(x) \)[/tex] is given by:
[tex]\[
f(x) = 4x - x^2
\][/tex]
2. Evaluate the Function at [tex]\( x = \frac{3}{2} \)[/tex]:
We need to substitute [tex]\( x = \frac{3}{2} \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f\left(\frac{3}{2}\right) = 4 \left(\frac{3}{2}\right) - \left(\frac{3}{2}\right)^2
\][/tex]
First, calculate [tex]\( 4 \left(\frac{3}{2}\right) \)[/tex]:
[tex]\[
4 \left(\frac{3}{2}\right) = 4 \times \frac{3}{2} = 2 \times 3 = 6
\][/tex]
Next, calculate [tex]\( \left(\frac{3}{2}\right)^2 \)[/tex]:
[tex]\[
\left(\frac{3}{2}\right)^2 = \frac{9}{4}
\][/tex]
Now, subtract [tex]\( \frac{9}{4} \)[/tex] from 6:
[tex]\[
6 - \frac{9}{4}
\][/tex]
Convert 6 to a fraction with the same denominator:
[tex]\[
6 = \frac{24}{4}
\][/tex]
So, the subtraction becomes:
[tex]\[
\frac{24}{4} - \frac{9}{4} = \frac{15}{4}
\][/tex]
Therefore:
[tex]\[
f\left(\frac{3}{2}\right) = \frac{15}{4} = 3.75
\][/tex]
3. Take the Square Root of the Result:
Now, we need to find the square root of [tex]\( 3.75 \)[/tex]:
[tex]\[
\sqrt{3.75} \approx 1.9364916731037085
\][/tex]
So, the detailed steps are:
- Calculate [tex]\( f\left(\frac{3}{2}\right) \)[/tex] which results in 3.75.
- Then, take the square root of 3.75 to get approximately 1.9364916731037085.
Thus, the final result is approximately [tex]\( 1.9364916731037085 \)[/tex].
