Question

Given f(x) = 3x- + 4x - 5 and g(x) = 2x + 9, find f(x) + g(x), f(x) - g(x), f(x) • g(x), and (x). Find [f ° gl(x) and [g° f](x) for each f(x) and g(x). 6. f(x) = 2× + 5 8(x) = 3+ x 7. f(x) = 2x - 3 g（x）=X3-2x

Answer 1

Explanation:

Let's perform the operations for the given functions:

1. f(x) + g(x):

f(x) = 2x + 5

g(x) = 3x + 1

(f + g)(x) = f(x) + g(x) = (2x + 5) + (3x + 1) = 5x + 6

2. f(x) - g(x):

f(x) = 2x + 5

g(x) = 3x + 1

(f - g)(x) = f(x) - g(x) = (2x + 5) - (3x + 1) = -x + 4

3. f(x) * g(x):

f(x) = 2x + 5

g(x) = 3x + 1

(f * g)(x) = f(x) * g(x) = (2x + 5)(3x + 1) = 6x^2 + 2x + 15x + 5 = 6x^2 + 17x + 5

4. f(g(x)):

f(x) = 2x + 5

g(x) = 3x - 2

f(g(x)) = f(3x - 2) = 2(3x - 2) + 5 = 6x - 4 + 5 = 6x + 1

5. g(f(x)):

f(x) = 2x + 5

g(x) = 3x - 2

g(f(x)) = g(2x + 5) = 3(2x + 5) - 2 = 6x + 15 - 2 = 6x + 13

For the functions f(x) = 2x - 3 and g(x) = x^3 - 2x:

6. f(x) + g(x):

(f + g)(x) = (2x - 3) + (x^3 - 2x) = x^3 - 2x + 2x - 3 = x^3 - 3

7. f(x) - g(x):

(f - g)(x) = (2x - 3) - (x^3 - 2x) = 2x - 3 - x^3 + 2x = -x^3 + 4x - 3

8. f(x) * g(x):

(f * g)(x) = (2x - 3)(x^3 - 2x) is the product of these two functions, which you can multiply if needed.

9. f(g(x)):

f(g(x)) = f(x^3 - 2x) is the composition of these two functions, and it can be computed as f(x^3 - 2x).

10. g(f(x)):

g(f(x)) = g(2x - 3) is the composition of these two functions, and it can be computed as g(2x - 3).