f(x)=x^2+3x+1,g(x)=2x-3, find fog and gog
fog
= f(g(x))
= f(2x-3)
= (2x-3)^2 + 3(2x-3) + 1
= 4x^2 - 12x + 9 + 6x - 9 + 1
= 4x^2 - 6x + 1
now do gog in the same way
Good
To find the composition of two functions f(x) and g(x), denoted as fog, we substitute g(x) into f(x) by replacing every instance of x in f(x) with g(x).
Given that f(x) = x^2 + 3x + 1 and g(x) = 2x - 3, we can find fog:
fog(x) = f(g(x))
Replacing x in f(x) with g(x):
fog(x) = f(g(x)) = (g(x))^2 + 3(g(x)) + 1
Substituting g(x) = 2x - 3:
fog(x) = (2x - 3)^2 + 3(2x - 3) + 1
Expanding and simplifying:
fog(x) = (4x^2 - 12x + 9) + (6x - 9) + 1
fog(x) = 4x^2 - 12x + 6x + 9 - 9 + 1
fog(x) = 4x^2 - 6x + 1
Therefore, fog(x) = 4x^2 - 6x + 1.
Similarly, we can find gog(x):
gog(x) = g(g(x))
Replacing x in g(x) with g(x):
gog(x) = g(g(x)) = 2(g(x)) - 3
Substituting g(x) = 2x - 3:
gog(x) = 2(2x - 3) - 3
Simplifying:
gog(x) = 4x - 6 - 3
gog(x) = 4x - 9
Therefore, gog(x) = 4x - 9.
To find the composition of two functions, we substitute the second function (g(x)) into the first function (f(x)). Let's find fog:
Step 1: Start with the function f(x) = x^2 + 3x + 1.
Step 2: Substitute g(x) = 2x - 3 into f(x):
f(g(x)) = (2x - 3)^2 + 3(2x - 3) + 1.
Let's simplify this expression:
Step 3: Expand (2x - 3)^2:
f(g(x)) = (4x^2 - 12x + 9) + 6x - 9 + 1.
f(g(x)) = 4x^2 - 12x + 6x + 9 - 9 + 1.
f(g(x)) = 4x^2 - 6x + 1.
So, fog = 4x^2 - 6x + 1.
Now, let's find gog:
Step 4: Substitute g(x) = 2x - 3 into itself:
g(g(x)) = 2(2x - 3) - 3.
Simplifying this expression:
Step 5: Distribute 2 to (2x - 3):
g(g(x)) = 4x - 6 - 3.
g(g(x)) = 4x - 9.
Therefore, gog = 4x - 9.
So, fog = 4x^2 - 6x + 1 and gog = 4x - 9.