a. Evaluate the following functions at [tex]\( x = -31 \)[/tex].

1. [tex]\( f(x) = x^3 - 64 \)[/tex]
2. [tex]\( g(x) = \left| x^3 - 3x^2 + 3x - 1 \right| \)[/tex]
3. [tex]\( r(x) = \sqrt{3 - 2x} \)[/tex]
4. [tex]\( q(x) = \frac{3x + 1}{x^2 + 7x + 10} \)[/tex]



Answer :

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]