Question 1:
For this case we have the following functions:
[tex]f (x) = 4x + 1\\g (x) = x ^ 3 + 1[/tex]
We must find[tex]g_ {o} f (0[/tex]):
By definition we have to:
[tex]g_ {o} f = g (f (x))\\f_ {o} g = f (g (x))[/tex]
[tex]g (f (x)) = (4x + 1) ^ 3 + 1[/tex]
We substitute [tex]x = 0[/tex]:
[tex]g (f (0)) = (4 (0) +1) ^ 3 + 1 = 1 ^ 3 + 1 = 2[/tex]
So, we have that [tex]g (f (0)) = 2[/tex]
Answer:
[tex]g (f (0)) = 2[/tex]
Question 2:
For this case we have the following functions:
[tex]f (x) = 4x + 1\\g (x) = x ^ 3 + 1[/tex]
We must find [tex]f_ {o} g (0)[/tex]:
By definition we have to:
[tex]f_ {o} g = f (g (x))\\f (g (x)) = 4 (x ^ 3 + 1) + 1 = 4x ^ 3 + 4 + 1 = 4x ^ 3 + 5[/tex]
We substitute [tex]x = 0[/tex]:
[tex]f (g (0)) = 4 (0) ^ 3 + 5 = 5[/tex]
Answer:
[tex]f (g (0)) = 5[/tex]
Question 3:
For this case we must find the inverse of the following function:
[tex]h (x) = \frac {2x + 1} {3}[/tex]
To do this, we follow the steps below:
We change y for [tex]h (x)[/tex]:
[tex]y = \frac {2x + 1} {3}[/tex]
We exchange variables:
[tex]x = \frac {2y + 1} {3}[/tex]
We clear the value of the variable "y":
[tex]3x = 2y + 1\\3x-1 = 2y\\y = \frac {3x} {2} - \frac {1} {2}[/tex]
We change y for [tex]h^{ -1} (x)[/tex]:
[tex]h ^{ - 1} (x) = \frac {3x} {2} - \frac {1} {2}[/tex]
Answer:
[tex]h ^ {- 1} (x) = \frac {3x} {2} - \frac {1} {2}[/tex]
