If f (x) = x² + 3 and g (x) = 3x - 1 then find the following:
1a. (f+g)(x)
1b. (f+g)(3)
1c. (f-g)(x)
1d. (fg)(x)
1e. (fg)(-1)
1a. (f+g)(x) = f(x)+g(x) = x^2 + 3 + 3x - 1 = x^2 + 3x + 2
1b. (f+g)(3) = 9+3+2 = 14
1c. (f-g) = x^2 - 3x + 4
1d. (fg)(x) = f(x)*g(x) = (x^2+3)(3x-1) = 3x^3 - x^2 + 9x - 3
1e. (fg)(-1) = -3 - 1 - 9 - 3 = -16
To find the given expressions, we will apply the operations of addition and multiplication to the given functions. Let's work through each part step by step:
1a. (f+g)(x) denotes the sum of the functions f(x) and g(x). To calculate this, we add the two functions together.
f(x) = x² + 3
g(x) = 3x - 1
Adding f(x) and g(x) together, we get:
(f+g)(x) = (x² + 3) + (3x - 1)
Combine like terms:
(f+g)(x) = x² + 3x + 2
Therefore, (f+g)(x) is equal to x² + 3x + 2.
1b. To find (f+g)(3), we substitute 3 for x in the expression we found in 1a.
(f+g)(x) = x² + 3x + 2
Substitute x = 3:
(f+g)(3) = (3)² + 3(3) + 2
= 9 + 9 + 2
= 20
Hence, (f+g)(3) equals 20.
1c. (f-g)(x) represents the difference between the functions f(x) and g(x). To calculate this, we subtract g(x) from f(x).
f(x) = x² + 3
g(x) = 3x - 1
Subtract g(x) from f(x):
(f-g)(x) = (x² + 3) - (3x - 1)
Remove parentheses and combine like terms:
(f-g)(x) = x² + 3 - 3x + 1
= x² - 3x + 4
Therefore, (f-g)(x) simplifies to x² - 3x + 4.
1d. (fg)(x) denotes the product of the functions f(x) and g(x). To calculate this, we multiply the two functions together.
f(x) = x² + 3
g(x) = 3x - 1
Multiply f(x) by g(x):
(fg)(x) = (x² + 3)(3x - 1)
Use the distributive property to combine like terms:
(fg)(x) = x²(3x) + x²(-1) + 3(3x) + 3(-1)
= 3x³ - x² + 9x - 3
Thus, (fg)(x) simplifies to 3x³ - x² + 9x - 3.
1e. To find (fg)(-1), we substitute -1 for x in the expression we found in 1d.
(fg)(x) = 3x³ - x² + 9x - 3
Substitute x = -1:
(fg)(-1) = 3(-1)³ - (-1)² + 9(-1) - 3
= -3 - 1 - 9 - 3
= -16
Hence, (fg)(-1) is equal to -16.