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Given the function [tex]f(x) = 4x - x^2[/tex], find [tex]\sqrt{f\left(\frac{3}{2}\right)}[/tex].

Sagot :

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].