For the function f(x)=5e^x and g(x)=x^5, find the following:
(a) f(g(1))=
(b) g(f(1))=
(c) f(g(x))=
(d) g(f(x))=
(e) f(t)g(t)=
simple substitution and manipulation
I will do b) and c)
b) f(1) = 5e^1 = 5e
then g(f(1)) = g(5e)
= (5e)^5 or 3125e^5
c)
f(g(x))
= f(x^5)
= 5e^(x^5)
You do the others, show me your work and I will check it.
a.) g(1)= 1^5= 1
f(g(1))=5e^1=
5e?
d.) g(f(x))= g(5e^x)
= (5e^x)^5?
e.) I'm really guessing on this last one because I'm not sure but I'll try it anyway
((5e)^t)(t^5)?
To find the values of these functions, we will substitute the given values into the functions and simplify the expressions.
(a) To find f(g(1)), we substitute x = 1 into g(x) first:
g(1) = 1^5 = 1
Now, substitute g(1) into f(x):
f(g(1)) = f(1) = 5e^1 = 5e
Therefore, f(g(1)) = 5e.
(b) To find g(f(1)), we substitute x = 1 into f(x):
f(1) = 5e^1 = 5e
Now, substitute f(1) into g(x):
g(f(1)) = g(5e) = (5e)^5 = 3125e^5
Therefore, g(f(1)) = 3125e^5.
(c) To find f(g(x)), substitute g(x) into f(x):
f(g(x)) = 5e^(g(x)) = 5e^(x^5)
Therefore, f(g(x)) = 5e^(x^5).
(d) To find g(f(x)), substitute f(x) into g(x):
g(f(x)) = (f(x))^5 = (5e^x)^5 = 3125e^(5x)
Therefore, g(f(x)) = 3125e^(5x).
(e) To find f(t)g(t), substitute t into f(t) and g(t) separately and then multiply the results:
f(t)g(t) = (5e^t)(t^5)
Therefore, f(t)g(t) = 5te^t * t^5 = 5t^6e^t
To solve these questions, we'll first find g(1) and f(1), then substitute those values into the functions f(x) and g(x). Finally, we'll perform the necessary operations to evaluate the expressions.
(a) To find f(g(1)), start by evaluating g(1):
g(1) = (1)^5 = 1
Substitute g(1) into f(x):
f(g(1)) = f(1) = 5e^1 = 5e
Therefore, f(g(1)) = 5e.
(b) To find g(f(1)), start by evaluating f(1):
f(1) = 5e^1 = 5e
Substitute f(1) into g(x):
g(f(1)) = g(5e) = (5e)^5 = 3125e^5
Therefore, g(f(1)) = 3125e^5.
(c) To find f(g(x)), substitute g(x) into f(x):
f(g(x)) = 5e^(g(x))
Therefore, f(g(x)) = 5e^(x^5).
(d) To find g(f(x)), substitute f(x) into g(x):
g(f(x)) = (f(x))^5 = (5e^x)^5 = 3125e^(5x)
Therefore, g(f(x)) = 3125e^(5x).
(e) To find f(t)g(t), multiply f(t) by g(t):
f(t)g(t) = (5e^t)*(t^5) = 5e^t * t^5
Therefore, f(t)g(t) = 5e^t * t^5.