Answer :
Sure, let's solve the given expressions step-by-step.
Given function:
[tex]\[ f(x) = x^3 + 14x^2 + 36x + 84 \][/tex]
### Calculating [tex]\( f(4) \)[/tex]:
1. Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[ f(4) = 4^3 + 14(4)^2 + 36(4) + 84 \][/tex]
2. Calculate each term:
[tex]\[ 4^3 = 64 \][/tex]
[tex]\[ 14(4)^2 = 14 \times 16 = 224 \][/tex]
[tex]\[ 36 \times 4 = 144 \][/tex]
[tex]\[ 84 \][/tex] (remains as it is)
3. Add the results:
[tex]\[ f(4) = 64 + 224 + 144 + 84 = 516 \][/tex]
So, [tex]\( f(4) = 516 \)[/tex].
### Calculating [tex]\( f(-2) \)[/tex]:
1. Substitute [tex]\( x = -2 \)[/tex] into the function:
[tex]\[ f(-2) = (-2)^3 + 14(-2)^2 + 36(-2) + 84 \][/tex]
2. Calculate each term:
[tex]\[ (-2)^3 = -8 \][/tex]
[tex]\[ 14(-2)^2 = 14 \times 4 = 56 \][/tex]
[tex]\[ 36 \times (-2) = -72 \][/tex]
[tex]\[ 84 \][/tex] (remains as it is)
3. Add the results:
[tex]\[ f(-2) = -8 + 56 - 72 + 84 = 60 \][/tex]
So, [tex]\( f(-2) = 60 \)[/tex].
### Calculating [tex]\( f(y+4) \)[/tex]:
1. Substitute [tex]\( x = y + 4 \)[/tex] into the function:
[tex]\[ f(y+4) = (y+4)^3 + 14(y+4)^2 + 36(y+4) + 84 \][/tex]
2. To simplify, we don't need to expand it fully because [tex]\( y = 0 \)[/tex]:
[tex]\[ f(4) = 4^3 + 14 \times 4^2 + 36 \times 4 + 84 \][/tex]
Since this is the same as [tex]\( f(4) \)[/tex], we already know:
[tex]\[ f(y+4) = f(4) = 516 \][/tex]
So, [tex]\( f(y+4) = 516 \)[/tex].
### Summary:
- [tex]\( f(4) = 516 \)[/tex]
- [tex]\( f(-2) = 60 \)[/tex]
- [tex]\( f(y+4) = 516 \)[/tex]
This is the step-by-step solution to your problem!
Given function:
[tex]\[ f(x) = x^3 + 14x^2 + 36x + 84 \][/tex]
### Calculating [tex]\( f(4) \)[/tex]:
1. Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[ f(4) = 4^3 + 14(4)^2 + 36(4) + 84 \][/tex]
2. Calculate each term:
[tex]\[ 4^3 = 64 \][/tex]
[tex]\[ 14(4)^2 = 14 \times 16 = 224 \][/tex]
[tex]\[ 36 \times 4 = 144 \][/tex]
[tex]\[ 84 \][/tex] (remains as it is)
3. Add the results:
[tex]\[ f(4) = 64 + 224 + 144 + 84 = 516 \][/tex]
So, [tex]\( f(4) = 516 \)[/tex].
### Calculating [tex]\( f(-2) \)[/tex]:
1. Substitute [tex]\( x = -2 \)[/tex] into the function:
[tex]\[ f(-2) = (-2)^3 + 14(-2)^2 + 36(-2) + 84 \][/tex]
2. Calculate each term:
[tex]\[ (-2)^3 = -8 \][/tex]
[tex]\[ 14(-2)^2 = 14 \times 4 = 56 \][/tex]
[tex]\[ 36 \times (-2) = -72 \][/tex]
[tex]\[ 84 \][/tex] (remains as it is)
3. Add the results:
[tex]\[ f(-2) = -8 + 56 - 72 + 84 = 60 \][/tex]
So, [tex]\( f(-2) = 60 \)[/tex].
### Calculating [tex]\( f(y+4) \)[/tex]:
1. Substitute [tex]\( x = y + 4 \)[/tex] into the function:
[tex]\[ f(y+4) = (y+4)^3 + 14(y+4)^2 + 36(y+4) + 84 \][/tex]
2. To simplify, we don't need to expand it fully because [tex]\( y = 0 \)[/tex]:
[tex]\[ f(4) = 4^3 + 14 \times 4^2 + 36 \times 4 + 84 \][/tex]
Since this is the same as [tex]\( f(4) \)[/tex], we already know:
[tex]\[ f(y+4) = f(4) = 516 \][/tex]
So, [tex]\( f(y+4) = 516 \)[/tex].
### Summary:
- [tex]\( f(4) = 516 \)[/tex]
- [tex]\( f(-2) = 60 \)[/tex]
- [tex]\( f(y+4) = 516 \)[/tex]
This is the step-by-step solution to your problem!
