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Find (fog)(2) and (f+g)(2) when f(x)= 1/x and g(x)=4x+9
To find (fog)(2), which represents the composition of functions f(x) and g(x), follow these steps:
1. Start by substituting the expression for g(x) into f(x). This is indicated by the notation "fog," which means f(g(x)).
So, (fog)(x) = f(g(x)) = f(4x + 9)
2. Next, substitute x = 2 into the expression for (fog)(x). This means you want to evaluate (fog)(2).
Therefore, (fog)(2) = f(4(2) + 9)
3. Simplify the expression inside f(x).
(fog)(2) = f(8 + 9) = f(17)
4. Now, substitute x = 17 into the expression for f(x):
(fog)(2) = f(17) = 1/17
So, (fog)(2) = 1/17.
To find (f+g)(2), follow these steps:
1. Start by adding the functions f(x) and g(x) together.
(f+g)(x) = f(x) + g(x) = 1/x + (4x + 9)
2. Next, substitute x = 2 into the expression for (f+g)(x). This means you want to evaluate (f+g)(2).
Therefore, (f+g)(2) = 1/2 + (4(2) + 9)
3. Simplify the expression inside (f+g)(2).
(f+g)(2) = 1/2 + (8 + 9) = 1/2 + 17 = 19/2
So, (f+g)(2) = 19/2.
To find (fog)(2) and (f+g)(2), we need to substitute the value of x=2 into each function separately and then perform the required operations.
1. To find (fog)(2):
- Start with the function g(x) = 4x + 9.
- Substitute x = 2: g(2) = 4(2) + 9 = 8 + 9 = 17.
- Now, we have the value of g(2) as 17.
- The function (fog)(x) represents the composition of f(x) and g(x), so we substitute g(2) into f(x) as follows:
f(g(2)) = f(17).
- For the function f(x) = 1/x, we substitute x = 17:
f(17) = 1/17.
- Therefore, (fog)(2) = f(g(2)) = f(17) = 1/17.
2. To find (f+g)(2):
- Start with the functions f(x) = 1/x and g(x) = 4x + 9.
- Substitute x = 2 into both functions:
f(2) = 1/2 and g(2) = 4(2) + 9 = 8 + 9 = 17.
- Now, we have the values of f(2) as 1/2 and g(2) as 17.
- The function (f+g)(x) represents the sum of f(x) and g(x), so we substitute the values we obtained above into (f+g)(x):
(f+g)(2) = f(2) + g(2) = 1/2 + 17.
- To add these fractions, they need a common denominator:
(f+g)(2) = (1/2) + (17/1) = (1/2) + (34/2) = 35/2.
- Therefore, (f+g)(2) = 35/2.
In summary:
- (fog)(2) = f(g(2)) = f(17) = 1/17.
- (f+g)(2) = f(2) + g(2) = 1/2 + 17 = 35/2.