Certainly! Let's solve the given problem step by step, based on the given piecewise function:
[tex]\[ f(x) = \begin{cases}
-x - 4, & \text{if } x \leq 5 \\
2x^2 - 7, & \text{if } 5 < x \leq 0
\end{cases} \][/tex]
### 1. Finding [tex]\( f(-2) \)[/tex]:
First, determine which piece of the function to use by checking the value of [tex]\( x = -2 \)[/tex] against the given conditions.
Since [tex]\( -2 \leq 5 \)[/tex], we use the first piece of the function:
[tex]\[ f(x) = -x - 4 \][/tex]
Now, substitute [tex]\( x = -2 \)[/tex] into the function:
[tex]\[ f(-2) = -(-2) - 4 \][/tex]
Simplify inside the parentheses:
[tex]\[ f(-2) = 2 - 4 \][/tex]
Finally, perform the subtraction:
[tex]\[ f(-2) = -2 \][/tex]
Therefore,
[tex]\[ f(-2) = -2 \][/tex]
### 2. Finding [tex]\( f(7) \)[/tex]:
Next, determine which piece of the function to use by checking the value of [tex]\( x = 7 \)[/tex] against the given conditions.
The value [tex]\( x = 7 \)[/tex] does not satisfy any of the given conditions:
- It does not satisfy [tex]\( x \leq 5 \)[/tex] because [tex]\( 7 > 5 \)[/tex].
- It does not satisfy [tex]\( 5 < x \leq 0 \)[/tex] because [tex]\( 7 > 5 \)[/tex] but [tex]\( 7 \)[/tex] is not less than or equal to [tex]\( 0 \)[/tex].
Since [tex]\( x = 7 \)[/tex] does not fall into any of the intervals provided in the piecewise function, [tex]\( f(7) \)[/tex] is not defined.
Thus,
[tex]\[ f(7) = \text{undefined} \][/tex]
To summarize:
[tex]\[ f(-2) = -2 \][/tex]
[tex]\[ f(7) \text{ is undefined} \][/tex]
