Example 1:
Let f(x) = ex and g(x) = �ãx
Find:
(a) f(g(x))
(b) g(f(x))
let f(x)= e^x and g(x)= sqaure root x
Thanks for rewriting the question. Some of the symbols could not be recognized.
f[g(x)] = e^x^1/2
g[f(x)] = sqrt e^x = e^(x/2)
so that is the answer for a and b
To find the compositions of the functions f(g(x)) and g(f(x)), you first need to understand how to compute function compositions.
(a) To find f(g(x)), you follow these steps:
1. Start with the function g(x), which is given as g(x) = �ãx.
2. Substitute x into the function f(x), which is given as f(x) = ex.
3. Replace x with the expression for g(x).
f(g(x)) = f(�ãx)
4. Substitute the expression in g(x) into the function f(x).
f(g(x)) = f(�ãx) = e(�ãx)
Therefore, f(g(x)) = e(�ãx).
(b) To find g(f(x)), you follow these steps:
1. Start with the function f(x), which is given as f(x) = ex.
2. Substitute x into the function g(x), which is given as g(x) = �ãx.
3. Replace x with the expression for f(x).
g(f(x)) = g(ex)
4. Substitute the expression in f(x) into the function g(x).
g(f(x)) = g(ex) = �ã(ex)
Therefore, g(f(x)) = �ã(ex).
Now you have found the compositions of the functions f(g(x)) and g(f(x)).