Sure, let's evaluate each function step-by-step at [tex]\( x = -31 \)[/tex].
1. Function: [tex]\( f(x) = x^3 - 64 \)[/tex]
- Substitute [tex]\( x = -31 \)[/tex]:
[tex]\[
f(-31) = (-31)^3 - 64
\][/tex]
- Compute [tex]\( (-31)^3 \)[/tex]:
[tex]\[
(-31)^3 = -31 \cdot -31 \cdot -31 = -29791 - 64
\][/tex]
- Subtract 64:
[tex]\[
-29791 - 64 = -29855
\][/tex]
Thus,
[tex]\[
f(-31) = -29855
\][/tex]
2. Function: [tex]\( g(x) = \left|x^3 - 3x^2 + 3x - 1\right| \)[/tex]
- Substitute [tex]\( x = -31 \)[/tex]:
[tex]\[
g(-31) = \left| (-31)^3 - 3(-31)^2 + 3(-31) - 1 \right|
\][/tex]
- Compute each term separately:
[tex]\[
(-31)^3 = -29791
\][/tex]
[tex]\[
3(-31)^2 = 3 \cdot 961 = 2883
\][/tex]
[tex]\[
3(-31) = -93
\][/tex]
- Combine these:
[tex]\[
-29791 - 2883 - 93 - 1 = -32768
\][/tex]
- Take the absolute value:
[tex]\[
\left| -32768 \right| = 32768
\][/tex]
Thus,
[tex]\[
g(-31) = 32768
\][/tex]
3. Function: [tex]\( r(x) = \sqrt{3 - 2x} \)[/tex]
- Substitute [tex]\( x = -31 \)[/tex]:
[tex]\[
r(-31) = \sqrt{3 - 2(-31)}
\][/tex]
- Simplify the expression inside the square root:
[tex]\[
3 + 62 = 65
\][/tex]
- Take the square root:
[tex]\[
\sqrt{65} = 8.06225774829855
\][/tex]
Thus,
[tex]\[
r(-31) = 8.06225774829855
\][/tex]
4. Function: [tex]\( q(x) = \frac{3x + 1}{x^2 + 7x + 10} \)[/tex]
- Substitute [tex]\( x = -31 \)[/tex]:
[tex]\[
q(-31) = \frac{3(-31) + 1}{(-31)^2 + 7(-31) + 10}
\][/tex]
- Compute the numerator:
[tex]\[
3(-31) + 1 = -93 + 1 = -92
\][/tex]
- Compute the denominator:
[tex]\[
(-31)^2 = 961
\][/tex]
[tex]\[
7(-31) = -217
\][/tex]
[tex]\[
961 - 217 + 10 = 754
\][/tex]
- Divide the numerator by the denominator:
[tex]\[
\frac{-92}{754} = -0.1220159151193634
\][/tex]
Thus,
[tex]\[
q(-31) = -0.1220159151193634
\][/tex]
In summary, the evaluations are:
[tex]\[
f(-31) = -29855
\][/tex]
[tex]\[
g(-31) = 32768
\][/tex]
[tex]\[
r(-31) = 8.06225774829855
\][/tex]
[tex]\[
q(-31) = -0.1220159151193634
\][/tex]
