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) = x + 5
g(x) = 5x^2+15x-3
just substitute f and g as needed
f(x) + g(x) = (x+5) + (5x^2 + 15x - 3)
= 5x^2 + 16x + 2
f(x) + g(x) = (x+5) - (5x^2 + 15x - 3)
= -5x^2 - 14x + 8
f(x) * g(x) = (x+5) * (5x^2 + 15x - 3)
= 5x^3 + 20x^2 + 72x - 15
f(x) + g(x) = (x+5) / (5x^2 + 15x - 3)
perils of copy/paste:
2nd and 4th lines should read f(x)-g(x) and f(x)/g(x)
To find the sum, difference, product, and quotient of two functions, f(x) and g(x), you need to perform the respective operations on their expressions. Let's go through each of the operations step by step for the given functions:
(a) f + g:
To find the sum, f + g, you simply add the expressions of f(x) and g(x):
f(x) + g(x) = (x + 5) + (5x^2 + 15x - 3)
Now, combine like terms:
f(x) + g(x) = 5x^2 + (x + 15x) + (5 - 3)
Combine the x terms and the constant terms:
f(x) + g(x) = 5x^2 + 16x + 2
Therefore, f + g = 5x^2 + 16x + 2.
(b) f - g:
To find the difference, f - g, you subtract the expression of g(x) from f(x):
f(x) - g(x) = (x + 5) - (5x^2 + 15x - 3)
Now, distribute the negative sign to g(x):
f(x) - g(x) = x + 5 - 5x^2 - 15x + 3
Rearrange the terms to have like terms together:
f(x) - g(x) = -5x^2 + (x - 15x) + (5 + 3)
Combine the x terms and the constant terms:
f(x) - g(x) = -5x^2 - 14x + 8
Therefore, f - g = -5x^2 - 14x + 8.
(c) f * g:
To find the product, f * g, you multiply the expressions of f(x) and g(x):
f(x) * g(x) = (x + 5) * (5x^2 + 15x - 3)
Now, use the distributive property to multiply each term:
f(x) * g(x) = x(5x^2 + 15x - 3) + 5(5x^2 + 15x - 3)
Multiply each term separately:
f(x) * g(x) = 5x^3 + 15x^2 - 3x + 25x^2 + 75x - 15
Combine like terms:
f(x) * g(x) = 5x^3 + (15x^2 + 25x^2) + (-3x + 75x) - 15
Simplify further:
f(x) * g(x) = 5x^3 + 40x^2 + 72x - 15
Therefore, f * g = 5x^3 + 40x^2 + 72x - 15.
(d) f / g:
To find the quotient, f / g, you divide the expression of f(x) by g(x). However, before that, we need to ensure that the denominator function is not equal to zero for any value of x.
Since g(x) = 5x^2 + 15x - 3 is not zero for any value of x, we can proceed.
f(x) / g(x) = (x + 5) / (5x^2 + 15x - 3)
To simplify this expression further and potentially find common factors to cancel, we cannot determine without additional instructions or restrictions on x.
Therefore, f / g = (x + 5) / (5x^2 + 15x - 3).