Given f(x) and g(x) below, find:
(a) f + g (b) f – g (c)f*g(d)f/g
Please show all of your work.
f(x) = 5x^2-5
g(x) = x – 7
What's the problem? Just substitute for f and g:
(c) f*g = (5x^2-5)(x-7) = 5(x-1)^2(x-7) = x^3 - 9x^2 + 15x - 7
To find the operations (a) f + g, (b) f - g, (c) f * g, and (d) f / g, we need to perform the respective operations on the functions f(x) and g(x).
(a) f + g:
To add two functions, we add the corresponding terms.
f(x) = 5x^2 - 5
g(x) = x - 7
To find f + g, replace f(x) with 5x^2 - 5 and g(x) with x - 7, and then add the terms.
(f + g)(x) = (5x^2 - 5) + (x - 7)
(f + g)(x) = 5x^2 + x - 5 - 7
(f + g)(x) = 5x^2 + x - 12
Therefore, f + g = 5x^2 + x - 12.
(b) f - g:
To subtract two functions, we subtract the corresponding terms.
(f - g)(x) = (5x^2 - 5) - (x - 7)
(f - g)(x) = 5x^2 - 5 - x + 7
(f - g)(x) = 5x^2 - x + 2
Therefore, f - g = 5x^2 - x + 2.
(c) f * g:
To multiply two functions, we use the distributive property.
(f * g)(x) = (5x^2 - 5)(x - 7)
(f * g)(x) = 5x^3 - 35x^2 - 5x + 35
Therefore, f * g = 5x^3 - 35x^2 - 5x + 35.
(d) f / g:
To divide two functions, we need to use long division. However, first, we need to check if g(x) is not equal to zero for all x values.
g(x) = x - 7
For g(x) to be nonzero:
x - 7 ≠ 0
x ≠ 7
Therefore, g(x) is not equal to zero for all x values, so we can proceed with division.
To find f / g, we perform long division between f(x) = 5x^2 - 5 and g(x) = x - 7.
5x - 12
____________________
x - 7 | 5x^2 + 0x - 5
- (5x^2 - 35x)
_________________
35x - 5
- (35x - 245)
_______________
240
Therefore, f / g = 5x - 12.
In summary:
(a) f + g = 5x^2 + x - 12
(b) f - g = 5x^2 - x + 2
(c) f * g = 5x^3 - 35x^2 - 5x + 35
(d) f / g = 5x - 12