Certainly! Let's break down the solution step by step.
1. Define the given functions:
- [tex]\( f(x) = [x - 2] \)[/tex], where [tex]\( [ \cdot ] \)[/tex] denotes the greatest integer function. This function returns the greatest integer less than or equal to [tex]\( x - 2 \)[/tex].
- [tex]\( g(x) = x^2 \)[/tex].
2. Calculate [tex]\( f\left( -\frac{1}{2} \right) \)[/tex]:
- Substitute [tex]\( x = -\frac{1}{2} \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f\left( -\frac{1}{2} \right) = \left[ -\frac{1}{2} - 2 \right]
\][/tex]
- Simplify inside the brackets:
[tex]\[
-\frac{1}{2} - 2 = -\frac{1}{2} - \frac{4}{2} = -\frac{5}{2}
\][/tex]
- The greatest integer less than or equal to [tex]\( -\frac{5}{2} \)[/tex] is [tex]\( -3 \)[/tex], thus:
[tex]\[
f\left( -\frac{1}{2} \right) = -3
\][/tex]
3. Calculate [tex]\( g(f\left( -\frac{1}{2} \right)) \)[/tex]:
- We know that [tex]\( f\left( -\frac{1}{2} \right) = -3 \)[/tex]. Now substitute [tex]\( x = -3 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(-3) = (-3)^2
\][/tex]
- Compute the square:
[tex]\[
(-3)^2 = 9
\][/tex]
4. Conclusion:
- The value of [tex]\( (g \circ f)\left( -\frac{1}{2} \right) \)[/tex] is [tex]\( 9 \)[/tex].
Thus, the correct answer is [tex]\(\boxed{9}\)[/tex].
