Let f(x)=2x2−7x+5 and g(x)=x2+9. Find f+g, f−g, f⋅g, and fg. Simplify your answers.
1. f+g=
2. f−g=
3. f⋅g=
4. f/g=
To find the values of f+g, f-g, f⋅g, and f/g, you need to perform the respective operations on the given functions f(x) and g(x). Let's solve each one step-by-step:
1. f+g:
To find f+g, you need to add the functions f(x) and g(x):
f(x) + g(x) = (2x^2 - 7x + 5) + (x^2 + 9)
Combine like terms:
= 2x^2 - 7x + 5 + x^2 + 9
Combine like terms again:
= 3x^2 - 7x + 14
So, f+g = 3x^2 - 7x + 14.
2. f-g:
Similarly, to find f-g, subtract the function g(x) from f(x):
f(x) - g(x) = (2x^2 - 7x + 5) - (x^2 + 9)
Combine like terms:
= 2x^2 - 7x + 5 - x^2 - 9
Combine like terms again:
= x^2 - 7x - 4
So, f-g = x^2 - 7x - 4.
3. f⋅g:
To find f⋅g, you need to multiply the functions f(x) and g(x):
f(x) ⋅ g(x) = (2x^2 - 7x + 5) ⋅ (x^2 + 9)
Apply the distributive property:
= 2x^2(x^2 + 9) - 7x(x^2 + 9) + 5(x^2 + 9)
Distribute:
= 2x^4 + 18x^2 - 7x^3 - 63x + 5x^2 + 45
Combine like terms:
= 2x^4 - 7x^3 + 23x^2 - 63x + 45
So, f⋅g = 2x^4 - 7x^3 + 23x^2 - 63x + 45.
4. f/g:
To find f/g, divide the function f(x) by g(x):
f(x) / g(x) = (2x^2 - 7x + 5) / (x^2 + 9)
Division of polynomials can be more complicated, so no further simplification can be done without additional instructions or restrictions.
So, f/g = (2x^2 - 7x + 5) / (x^2 + 9).