f(x)= 3x-5 and g(x)= 2-x^2 evaluate:
1.(fog)(x)
2.(gog)(-2)
(fog)(x) means you substitute the whole function of g(x) into the f(x),, we can actually rewrite this (fog)(x) as f(g(x)) --- i think this one is clearer.
now let's substitute g(x) to f(x):
(fog)(x) = f(g(x)) = 3x-5 = 3[g(x)] - 5
(fog)(x) = 3(2 - x^2) - 5
(fog)(x) = 6 - 3x^2 - 5
(fog)(x) = -3x^2 + 1
for #2, we do the same,, we substitute g(x) to g(x) then we evaluate the simplified expression at x=-2:
(gog)(x) = g(g(x)) = 2 - (g(x))^2
(gog)(x) = 2 - (2 - x^2)^2
(gog)(-2) = 2 - [2 - (-2)^2]^2
(gog)(-2) = 2 - [2 - 4]^2
(gog)(-2) = 2 - (-2)^2
(gog)(-2) = 2 - 4
(gog)(-2) = -2
hope this helps~ :)
Thanks a bunch
Yes thank you very much!
To evaluate the compositions of functions (f∘g)(x) and (g∘g)(-2), we need to substitute the given functions f(x) and g(x) into each other.
1. Evaluating (f∘g)(x):
To find (f∘g)(x), we need to substitute g(x) into f(x):
(f∘g)(x) = f(g(x))
First, let's find g(x):
g(x) = 2 - x^2
Now, substitute g(x) into f(x):
(f∘g)(x) = f(2-x^2)
Replacing x in f(x) with (2-x^2):
(f∘g)(x) = 3(2-x^2) - 5
Expanding the expression:
(f∘g)(x) = 6 - 3x^2 - 5
Combining like terms:
(f∘g)(x) = -3x^2 + 1
2. Evaluating (g∘g)(-2):
To find (g∘g)(-2), we need to substitute g(x) into g(x):
(g∘g)(x) = g(g(x))
Given g(x) = 2 - x^2, substitute it into itself:
(g∘g)(x) = g(2 - x^2)
Replacing x with -2 in g(x):
(g∘g)(x) = g(2 - (-2)^2)
Simplifying the exponent and computing the expression inside g():
(g∘g)(x) = g(2 - 4)
Now, we find the value of g(x) at 2 - 4:
(g∘g)(x) = g(-2)
Substitute -2 into g(x):
(g∘g)(x) = 2 - (-2)^2
Squaring -2:
(g∘g)(x) = 2 - 4
Calculating the expression:
(g∘g)(x) = -2
Therefore, (f∘g)(x) = -3x^2 + 1, and (g∘g)(-2) = -2.