Certainly! Let's work through the problem step-by-step.
We are given the following functions:
[tex]\[ f(x) = 3x + p \][/tex]
[tex]\[ g(x) = px + 4 \][/tex]
We also know that:
[tex]\[ f(g(x)) = 6x + q \][/tex]
First, let’s determine [tex]\( f(g(x)) \)[/tex].
1. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
First, we find [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = px + 4 \][/tex]
Now substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(px + 4) \][/tex]
2. Evaluate [tex]\( f(px + 4) \)[/tex]:
Using the expression for [tex]\( f(x) \)[/tex]:
[tex]\[ f(px + 4) = 3(px + 4) + p \][/tex]
3. Expand and simplify:
[tex]\[ f(px + 4) = 3(px) + 3(4) + p \][/tex]
[tex]\[ f(px + 4) = 3px + 12 + p \][/tex]
4. Set this equal to the given expression for [tex]\( f(g(x)) \)[/tex]:
We know from the problem statement:
[tex]\[ f(g(x)) = 6x + q \][/tex]
Therefore,
[tex]\[ 3px + 12 + p = 6x + q \][/tex]
5. Equate the coefficients of [tex]\( x \)[/tex] and the constant terms:
By comparing the coefficients of [tex]\( x \)[/tex] on both sides of the equation,
[tex]\[ 3p = 6 \][/tex]
Solving for [tex]\( p \)[/tex]:
[tex]\[ p = \frac{6}{3} \][/tex]
[tex]\[ p = 2 \][/tex]
6. Substitute [tex]\( p = 2 \)[/tex] into the constant term equation:
Now we substitute back into the equation for the constants,
[tex]\[ 12 + p = q \][/tex]
Substitute [tex]\( p = 2 \)[/tex]:
[tex]\[ 12 + 2 = q \][/tex]
[tex]\[ q = 14 \][/tex]
Therefore, the values of the constants are:
[tex]\[ p = 2 \][/tex]
[tex]\[ q = 14 \][/tex]
This completes the step-by-step solution to the problem.
