Answer :
To evaluate the piecewise function [tex]\( f(x) \)[/tex] at the given values [tex]\( f(-5), f(-3) \)[/tex], and [tex]\( f(2) \)[/tex], we need to consider the definition of [tex]\( f(x) \)[/tex] based on the value of [tex]\( x \)[/tex]:
[tex]\[ f(x)=\left\{\begin{array}{ll} -3 x^2-7 x+11 & \text{if } x \leq -3 \\ x^3+7 x-4 & \text{if } x > -3 \end{array}\right. \][/tex]
Let's proceed with the evaluations:
### Evaluation of [tex]\( f(-5) \)[/tex]
Since [tex]\( -5 \leq -3 \)[/tex], we use the first piece of the function:
[tex]\[ f(x) = -3 x^2 - 7 x + 11. \][/tex]
Substitute [tex]\( x = -5 \)[/tex] into the function:
[tex]\[ f(-5) = -3(-5)^2 - 7(-5) + 11. \][/tex]
Evaluating the expression:
[tex]\[ -3(-5)^2 = -3 \cdot 25 = -75, \][/tex]
[tex]\[ -7(-5) = 35, \][/tex]
[tex]\[ f(-5) = -75 + 35 + 11 = -29. \][/tex]
So, [tex]\( f(-5) = -29 \)[/tex].
### Evaluation of [tex]\( f(-3) \)[/tex]
Since [tex]\( -3 \leq -3 \)[/tex], we use the first piece of the function:
[tex]\[ f(x) = -3 x^2 - 7 x + 11. \][/tex]
Substitute [tex]\( x = -3 \)[/tex] into the function:
[tex]\[ f(-3) = -3(-3)^2 - 7(-3) + 11. \][/tex]
Evaluating the expression:
[tex]\[ -3(-3)^2 = -3 \cdot 9 = -27, \][/tex]
[tex]\[ -7(-3) = 21, \][/tex]
[tex]\[ f(-3) = -27 + 21 + 11 = 5. \][/tex]
So, [tex]\( f(-3) = 5 \)[/tex].
### Evaluation of [tex]\( f(2) \)[/tex]
Since [tex]\( 2 > -3 \)[/tex], we use the second piece of the function:
[tex]\[ f(x) = x^3 + 7 x - 4. \][/tex]
Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ f(2) = 2^3 + 7(2) - 4. \][/tex]
Evaluating the expression:
[tex]\[ 2^3 = 8, \][/tex]
[tex]\[ 7 \cdot 2 = 14, \][/tex]
[tex]\[ f(2) = 8 + 14 - 4 = 18. \][/tex]
So, [tex]\( f(2) = 18 \)[/tex].
### Summary of Evaluations
To summarize, the evaluations of the piecewise function [tex]\( f(x) \)[/tex] at the given points are:
[tex]\[ f(-5) = -29, \][/tex]
[tex]\[ f(-3) = 5, \][/tex]
[tex]\[ f(2) = 18. \][/tex]
[tex]\[ f(x)=\left\{\begin{array}{ll} -3 x^2-7 x+11 & \text{if } x \leq -3 \\ x^3+7 x-4 & \text{if } x > -3 \end{array}\right. \][/tex]
Let's proceed with the evaluations:
### Evaluation of [tex]\( f(-5) \)[/tex]
Since [tex]\( -5 \leq -3 \)[/tex], we use the first piece of the function:
[tex]\[ f(x) = -3 x^2 - 7 x + 11. \][/tex]
Substitute [tex]\( x = -5 \)[/tex] into the function:
[tex]\[ f(-5) = -3(-5)^2 - 7(-5) + 11. \][/tex]
Evaluating the expression:
[tex]\[ -3(-5)^2 = -3 \cdot 25 = -75, \][/tex]
[tex]\[ -7(-5) = 35, \][/tex]
[tex]\[ f(-5) = -75 + 35 + 11 = -29. \][/tex]
So, [tex]\( f(-5) = -29 \)[/tex].
### Evaluation of [tex]\( f(-3) \)[/tex]
Since [tex]\( -3 \leq -3 \)[/tex], we use the first piece of the function:
[tex]\[ f(x) = -3 x^2 - 7 x + 11. \][/tex]
Substitute [tex]\( x = -3 \)[/tex] into the function:
[tex]\[ f(-3) = -3(-3)^2 - 7(-3) + 11. \][/tex]
Evaluating the expression:
[tex]\[ -3(-3)^2 = -3 \cdot 9 = -27, \][/tex]
[tex]\[ -7(-3) = 21, \][/tex]
[tex]\[ f(-3) = -27 + 21 + 11 = 5. \][/tex]
So, [tex]\( f(-3) = 5 \)[/tex].
### Evaluation of [tex]\( f(2) \)[/tex]
Since [tex]\( 2 > -3 \)[/tex], we use the second piece of the function:
[tex]\[ f(x) = x^3 + 7 x - 4. \][/tex]
Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ f(2) = 2^3 + 7(2) - 4. \][/tex]
Evaluating the expression:
[tex]\[ 2^3 = 8, \][/tex]
[tex]\[ 7 \cdot 2 = 14, \][/tex]
[tex]\[ f(2) = 8 + 14 - 4 = 18. \][/tex]
So, [tex]\( f(2) = 18 \)[/tex].
### Summary of Evaluations
To summarize, the evaluations of the piecewise function [tex]\( f(x) \)[/tex] at the given points are:
[tex]\[ f(-5) = -29, \][/tex]
[tex]\[ f(-3) = 5, \][/tex]
[tex]\[ f(2) = 18. \][/tex]
