To determine [tex]\( f(-3) \)[/tex] for the piecewise function
[tex]\[ f(x)=\left\{
\begin{array}{cc}
x^3, & x<-3 \\
2 x^2-9, & -3 \leq x<4 \\
5 x+4, & x \geq 4
\end{array}
\right. \][/tex]
we need to identify which part of the piecewise definition applies to [tex]\( x = -3 \)[/tex].
1. First, we examine the conditions for each piece of the function that determine how [tex]\( f(x) \)[/tex] is defined:
- For [tex]\( x < -3 \)[/tex], [tex]\( f(x) = x^3 \)[/tex].
- For [tex]\( -3 \leq x < 4 \)[/tex], [tex]\( f(x) = 2x^2 - 9 \)[/tex].
- For [tex]\( x \geq 4 \)[/tex], [tex]\( f(x) = 5x + 4 \)[/tex].
2. Since [tex]\( -3 \leq -3 < 4 \)[/tex], [tex]\( x = -3 \)[/tex] falls within the second piece of the function: [tex]\( 2x^2 - 9 \)[/tex].
3. Now, we substitute [tex]\( x = -3 \)[/tex] into the function that applies to this interval:
[tex]\[ f(-3) = 2(-3)^2 - 9 \][/tex]
4. Calculate the expression inside the brackets first:
[tex]\[ (-3)^2 = 9 \][/tex]
5. Multiply by 2:
[tex]\[ 2 \times 9 = 18 \][/tex]
6. Subtract 9 from 18:
[tex]\[ 18 - 9 = 9 \][/tex]
Therefore, the value of [tex]\( f(-3) \)[/tex] is [tex]\( 9 \)[/tex].
Thus, [tex]\( f(-3) = 9 \)[/tex].
