Answer :
Let's solve each part of the question step-by-step.
1) Finding [tex]\((f-g)(x)\)[/tex]:
Given:
- [tex]\( f(x) = x^2 + 5x \)[/tex]
- [tex]\( g(x) = 4x - 1 \)[/tex]
To find [tex]\((f-g)(x)\)[/tex], subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex]:
[tex]\[
(f-g)(x) = f(x) - g(x)
= (x^2 + 5x) - (4x - 1)
= x^2 + 5x - 4x + 1
= x^2 + x + 1
\][/tex]
The correct expression is [tex]\(x^2 + x + 1\)[/tex].
2) Finding [tex]\((f \cdot g)(x)\)[/tex]:
Using the same functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x)
= (x^2 + 5x) \cdot (4x - 1)
\][/tex]
Now, apply the distributive property:
[tex]\[
(x^2 + 5x) \cdot (4x - 1) = x^2 \cdot 4x + x^2 \cdot (-1) + 5x \cdot 4x + 5x \cdot (-1)
\][/tex]
[tex]\[
= 4x^3 - x^2 + 20x^2 - 5x
\][/tex]
Combine like terms:
[tex]\[
= 4x^3 + 19x^2 - 5x
\][/tex]
The correct expression is [tex]\(4x^3 + 19x^2 - 5x\)[/tex].
3) Calculating [tex]\((f+g)(-1)\)[/tex]:
We need to evaluate [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] at [tex]\(x = -1\)[/tex]:
[tex]\[
f(-1) = (-1)^2 + 5(-1) = 1 - 5 = -4
\][/tex]
[tex]\[
g(-1) = 4(-1) - 1 = -4 - 1 = -5
\][/tex]
Now add these results:
[tex]\[
(f+g)(-1) = f(-1) + g(-1) = -4 + (-5) = -9
\][/tex]
The answer for [tex]\((f+g)(-1)\)[/tex] is [tex]\(-9\)[/tex].
In summary:
1) [tex]\((f-g)(x) = x^2 + x + 1\)[/tex]
2) [tex]\((f \cdot g)(x) = 4x^3 + 19x^2 - 5x\)[/tex]
3) [tex]\((f+g)(-1) = -9\)[/tex]
1) Finding [tex]\((f-g)(x)\)[/tex]:
Given:
- [tex]\( f(x) = x^2 + 5x \)[/tex]
- [tex]\( g(x) = 4x - 1 \)[/tex]
To find [tex]\((f-g)(x)\)[/tex], subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex]:
[tex]\[
(f-g)(x) = f(x) - g(x)
= (x^2 + 5x) - (4x - 1)
= x^2 + 5x - 4x + 1
= x^2 + x + 1
\][/tex]
The correct expression is [tex]\(x^2 + x + 1\)[/tex].
2) Finding [tex]\((f \cdot g)(x)\)[/tex]:
Using the same functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x)
= (x^2 + 5x) \cdot (4x - 1)
\][/tex]
Now, apply the distributive property:
[tex]\[
(x^2 + 5x) \cdot (4x - 1) = x^2 \cdot 4x + x^2 \cdot (-1) + 5x \cdot 4x + 5x \cdot (-1)
\][/tex]
[tex]\[
= 4x^3 - x^2 + 20x^2 - 5x
\][/tex]
Combine like terms:
[tex]\[
= 4x^3 + 19x^2 - 5x
\][/tex]
The correct expression is [tex]\(4x^3 + 19x^2 - 5x\)[/tex].
3) Calculating [tex]\((f+g)(-1)\)[/tex]:
We need to evaluate [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] at [tex]\(x = -1\)[/tex]:
[tex]\[
f(-1) = (-1)^2 + 5(-1) = 1 - 5 = -4
\][/tex]
[tex]\[
g(-1) = 4(-1) - 1 = -4 - 1 = -5
\][/tex]
Now add these results:
[tex]\[
(f+g)(-1) = f(-1) + g(-1) = -4 + (-5) = -9
\][/tex]
The answer for [tex]\((f+g)(-1)\)[/tex] is [tex]\(-9\)[/tex].
In summary:
1) [tex]\((f-g)(x) = x^2 + x + 1\)[/tex]
2) [tex]\((f \cdot g)(x) = 4x^3 + 19x^2 - 5x\)[/tex]
3) [tex]\((f+g)(-1) = -9\)[/tex]
