Answered

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Given that for all values of [tex][tex]$x$[/tex][/tex],

[tex]\[
\begin{array}{l}
f(x) = 3x + p \\
g(x) = px + 4
\end{array}
\][/tex]

and [tex][tex]$\operatorname{fg}(x) = 6x + q$[/tex][/tex],

where [tex][tex]$p$[/tex][/tex] and [tex][tex]$q$[/tex][/tex] are constants,

work out the values of [tex][tex]$p$[/tex][/tex] and [tex][tex]$q$[/tex][/tex].

Sagot :

GinnyAnswer
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.
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