Let f(x) = x2-x-6/x
g(x) = x-3, find
(f+g)(x)
(f-g)(x)
(f/g)(x)
Help !!! thanks
To find (f+g)(x), you need to add the functions f(x) and g(x) together. Here's how you can do that:
Step 1: Write down the functions f(x) and g(x):
- f(x) = (x^2 - x - 6) / x
- g(x) = x - 3
Step 2: Write out (f+g)(x) by substituting the functions f(x) and g(x) into the equation:
- (f+g)(x) = f(x) + g(x)
Step 3: Replace f(x) and g(x) in the equation with their respective expressions:
- (f+g)(x) = [(x^2 - x - 6) / x] + (x - 3)
Step 4: Find a common denominator for the fractions, in this case, it is x:
- (f+g)(x) = [(x^2 - x - 6) + x(x - 3)] / x
Step 5: Simplify the expression by combining like terms:
- (f+g)(x) = [(x^2 - x - 6 + x^2 - 3x)] / x
- (f+g)(x) = (2x^2 - 4x - 6) / x
To find (f-g)(x), you subtract the functions f(x) - g(x). The process is similar to finding (f+g)(x):
Step 1: Write down the functions f(x) and g(x):
- f(x) = (x^2 - x - 6) / x
- g(x) = x - 3
Step 2: Write out (f-g)(x) by substituting the functions f(x) and g(x) into the equation:
- (f-g)(x) = f(x) - g(x)
Step 3: Replace f(x) and g(x) in the equation with their respective expressions:
- (f-g)(x) = [(x^2 - x - 6) / x] - (x - 3)
Step 4: Find a common denominator for the fractions, in this case, it is x:
- (f-g)(x) = [(x^2 - x - 6) - x(x - 3)] / x
Step 5: Simplify the expression by combining like terms:
- (f-g)(x) = [(x^2 - x - 6 - x^2 + 3x)] / x
- (f-g)(x) = (4x - 6) / x
To find (f/g)(x), you divide the function f(x) by g(x). Here's how to do it:
Step 1: Write down the functions f(x) and g(x):
- f(x) = (x^2 - x - 6) / x
- g(x) = x - 3
Step 2: Write out (f/g)(x) by substituting the functions f(x) and g(x) into the equation:
- (f/g)(x) = f(x) / g(x)
Step 3: Replace f(x) and g(x) in the equation with their respective expressions:
- (f/g)(x) = [(x^2 - x - 6) / x] / (x - 3)
Step 4: Simplify the expression by multiplying the numerator by the reciprocal of the denominator:
- (f/g)(x) = (x^2 - x - 6) / x * 1 / (x - 3)
Step 5: Simplify further by multiplying the numerators and denominators:
- (f/g)(x) = (x^2 - x - 6) / (x(x - 3))
So, the functions (f+g)(x), (f-g)(x), and (f/g)(x) can be written as:
- (f+g)(x) = (2x^2 - 4x - 6) / x
- (f-g)(x) = (4x - 6) / x
- (f/g)(x) = (x^2 - x - 6) / (x(x - 3))