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Given the function

[tex]\[
f(x) = \left\{
\begin{array}{ll}
x^2 + 10 & \text{if } x \ \textless \ -1 \\
2x^2 - 3 & \text{if } -1 \leq x \leq 1 \\
9 & \text{if } x \ \textgreater \ 1
\end{array}
\right.
\][/tex]

Calculate the following values:

[tex]\[
\begin{array}{l}
f(-2) = \\
f(1) = \\
f(2) =
\end{array}
\][/tex]



Answer :

GinnyAnswer
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.

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