Certainly! Let's tackle each problem step-by-step.
### Problem 1: Given [tex]\( f(x) = 3x - 1 \)[/tex] and [tex]\( g(x) = 2x + 3 \)[/tex]
We are asked to find the composition of functions [tex]\( f(g(x)) \)[/tex].
1. Start with the expression for [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = 2x + 3
\][/tex]
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]. This means we replace every occurrence of [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( g(x) \)[/tex].
So, [tex]\( f(g(x)) \)[/tex] becomes:
[tex]\[
f(g(x)) = f(2x + 3)
\][/tex]
3. Now, substitute [tex]\( 2x + 3 \)[/tex] into [tex]\( f(x) = 3x - 1 \)[/tex]:
[tex]\[
f(2x + 3) = 3(2x + 3) - 1
\][/tex]
4. Distribute and simplify:
[tex]\[
f(2x + 3) = 3 \cdot 2x + 3 \cdot 3 - 1
\][/tex]
[tex]\[
f(2x + 3) = 6x + 9 - 1
\][/tex]
[tex]\[
f(2x + 3) = 6x + 8
\][/tex]
Thus, the composition [tex]\( f(g(x)) = 6x + 8 \)[/tex].
### Problem 2: Given [tex]\( v(x) = x + 7 \)[/tex] and [tex]\( p(x) = x + 10 \)[/tex]
We are asked to find the composition of functions [tex]\( v(p(x)) \)[/tex].
1. Start with the expression for [tex]\( p(x) \)[/tex]:
[tex]\[
p(x) = x + 10
\][/tex]
2. Substitute [tex]\( p(x) \)[/tex] into [tex]\( v(x) \)[/tex]. This means we replace every occurrence of [tex]\( x \)[/tex] in [tex]\( v(x) \)[/tex] with [tex]\( p(x) \)[/tex].
So, [tex]\( v(p(x)) \)[/tex] becomes:
[tex]\[
v(p(x)) = v(x + 10)
\][/tex]
3. Now, substitute [tex]\( x + 10 \)[/tex] into [tex]\( v(x) = x + 7 \)[/tex]:
[tex]\[
v(x + 10) = (x + 10) + 7
\][/tex]
4. Simplify the expression:
[tex]\[
v(x + 10) = x + 10 + 7
\][/tex]
[tex]\[
v(x + 10) = x + 17
\][/tex]
Thus, the composition [tex]\( v(p(x)) = x + 17 \)[/tex].
To sum up, the results of the given problems are:
1. [tex]\( f(g(x)) = 6x + 8 \)[/tex]
2. [tex]\( v(p(x)) = x + 17 \)[/tex]
