Sure! Let's calculate the values of the given function [tex]\( f(x) \)[/tex] for the specified inputs step-by-step.
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[
f(x)=\left\{
\begin{array}{ll}
x^2 + 10 & \text{if } x < -1 \\
2 x^2 - 3 & \text{if } -1 \leq x \leq 1 \\
9 & \text{if } x > 1
\end{array}
\right.
\][/tex]
### Calculation for [tex]\( f(-2) \)[/tex]:
When [tex]\( x = -2 \)[/tex]:
- According to the definition, for [tex]\( x < -1 \)[/tex], [tex]\( f(x) = x^2 + 10 \)[/tex].
- Substitute [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = (-2)^2 + 10 = 4 + 10 = 14
\][/tex]
So, [tex]\( f(-2) = 14 \)[/tex].
### Calculation for [tex]\( f(1) \)[/tex]:
When [tex]\( x = 1 \)[/tex]:
- According to the definition, for [tex]\( -1 \leq x \leq 1 \)[/tex], [tex]\( f(x) = 2 x^2 - 3 \)[/tex].
- Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2 \cdot (1)^2 - 3 = 2 \cdot 1 - 3 = 2 - 3 = -1
\][/tex]
So, [tex]\( f(1) = -1 \)[/tex].
### Calculation for [tex]\( f(2) \)[/tex]:
When [tex]\( x = 2 \)[/tex]:
- According to the definition, for [tex]\( x > 1 \)[/tex], [tex]\( f(x) = 9 \)[/tex].
So, [tex]\( f(2) = 9 \)[/tex].
### Summary
The values are:
[tex]\[
\begin{array}{l}
f(-2) = 14 \\
f(1) = -1 \\
f(2) = 9
\end{array}
\][/tex]
These calculations provide the function values for the given inputs.
