Sure, let's break down each part of the problem and solve it step by step:
### Part 4: Finding [tex]\( h(2) \)[/tex]
Given the function [tex]\( h(x) = x^2 - 5x + 7 \)[/tex], we need to find [tex]\( h(2) \)[/tex]:
1. Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ h(2) = 2^2 - 5(2) + 7 \][/tex]
2. Calculate the squared term and the product:
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ 5(2) = 10 \][/tex]
3. Substitute these values back into the equation:
[tex]\[ h(2) = 4 - 10 + 7 \][/tex]
4. Perform the arithmetic:
[tex]\[ 4 - 10 = -6 \][/tex]
[tex]\[ -6 + 7 = 1 \][/tex]
Thus, [tex]\( h(2) = 1 \)[/tex].
### Part 5: Finding [tex]\( h(-5) \)[/tex]
Now, we need to find [tex]\( h(-5) \)[/tex]:
1. Substitute [tex]\( x = -5 \)[/tex] into the function:
[tex]\[ h(-5) = (-5)^2 - 5(-5) + 7 \][/tex]
2. Calculate the squared term and the product:
[tex]\[ (-5)^2 = 25 \][/tex]
[tex]\[ 5(-5) = -25 \][/tex]
3. Substitute these values back into the equation:
[tex]\[ h(-5) = 25 + 25 + 7 \][/tex]
4. Perform the arithmetic:
[tex]\[ 25 + 25 = 50 \][/tex]
[tex]\[ 50 + 7 = 57 \][/tex]
Thus, [tex]\( h(-5) = 57 \)[/tex].
### Part 6: Finding [tex]\(\frac{1}{3} \cdot h(-8)\)[/tex]
We need to find [tex]\( \frac{1}{3} \cdot h(-8) \)[/tex]:
1. Substitute [tex]\( x = -8 \)[/tex] into the function:
[tex]\[ h(-8) = (-8)^2 - 5(-8) + 7 \][/tex]
2. Calculate the squared term and the product:
[tex]\[ (-8)^2 = 64 \][/tex]
[tex]\[ 5(-8) = -40 \][/tex]
3. Substitute these values back into the equation:
[tex]\[ h(-8) = 64 + 40 + 7 \][/tex]
4. Perform the arithmetic:
[tex]\[ 64 + 40 = 104 \][/tex]
[tex]\[ 104 + 7 = 111 \][/tex]
5. Calculate [tex]\(\frac{1}{3} \cdot 111 \)[/tex]:
[tex]\[ \frac{1}{3} \cdot 111 = 37 \][/tex]
Thus, [tex]\( \frac{1}{3} \cdot h(-8) = 37.0 \)[/tex].
### Part 9: Finding [tex]\( -5 \cdot g(4) -1 \)[/tex]
Given the function [tex]\( g(x) = 1 - \frac{3}{4}x \)[/tex], we need to find [tex]\( -5 \cdot g(4) - 1 \)[/tex]:
1. Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[ g(4) = 1 - \frac{3}{4}(4) \][/tex]
2. Calculate the product:
[tex]\[ \frac{3}{4}(4) = 3 \][/tex]
3. Substitute this value back into the equation:
[tex]\[ g(4) = 1 - 3 \][/tex]
4. Perform the arithmetic:
[tex]\[ 1 - 3 = -2 \][/tex]
5. Calculate [tex]\( -5 \cdot g(4) \)[/tex]:
[tex]\[ -5 \cdot (-2) = 10 \][/tex]
6. Subtract 1:
[tex]\[ 10 - 1 = 9 \][/tex]
Thus, [tex]\( -5 \cdot g(4) - 1 = 9.0 \)[/tex].
In summary:
- [tex]\( h(2) = 1 \)[/tex]
- [tex]\( h(-5) = 57 \)[/tex]
- [tex]\( \frac{1}{3} \cdot h(-8) = 37.0 \)[/tex]
- [tex]\( -5 \cdot g(4) -1 = 9.0 \)[/tex]
