Bartlett Fowler
10/21/2023 · Junior High School
Given that \( f(x)=p x+q \) and \( f^{3}(x)=8 x+14 \). (a) Cari nilai \( p \) dan nilai \( q \). Find the values of \( p \) and of \( q \).
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Step-by-step Solution
To find the values of \( p \) and \( q \) given that \( f(x) = px + q \) and \( f^3(x) = 8x + 14 \), we first need to understand what \( f^3(x) \) means. It represents the function \( f \) applied three times in succession.
1. **Calculate \( f^2(x) \)**:
\[
f^2(x) = f(f(x)) = f(px + q)
\]
Substituting \( px + q \) into \( f \):
\[
f(px + q) = p(px + q) + q = p^2x + pq + q
\]
Thus, we have:
\[
f^2(x) = p^2x + (pq + q)
\]
2. **Calculate \( f^3(x) \)**:
\[
f^3(x) = f(f^2(x)) = f(p^2x + (pq + q))
\]
Substituting \( p^2x + (pq + q) \) into \( f \):
\[
f(p^2x + (pq + q)) = p(p^2x + (pq + q)) + q = p^3x + p(pq + q) + q
\]
Simplifying this gives:
\[
f^3(x) = p^3x + (p^2q + pq + q)
\]
3. **Set \( f^3(x) \) equal to \( 8x + 14 \)**:
From the problem statement, we know:
\[
f^3(x) = 8x + 14
\]
Therefore, we can equate coefficients:
\[
p^3 = 8 \quad \text{(coefficient of } x\text{)}
\]
\[
p^2q + pq + q = 14 \quad \text{(constant term)}
\]
4. **Solve for \( p \)**:
From \( p^3 = 8 \), we find:
\[
p = 2
\]
5. **Substitute \( p \) into the second equation**:
Now substituting \( p = 2 \) into the second equation:
\[
p^2q + pq + q = 14
\]
becomes:
\[
2^2q + 2q + q = 14
\]
Simplifying this:
\[
4q + 2q + q = 14
\]
\[
7q = 14
\]
\[
q = 2
\]
6. **Final values**:
Thus, the values of \( p \) and \( q \) are:
\[
p = 2, \quad q = 2
\]
In conclusion, the values are:
\[
\boxed{2} \text{ for } p \text{ and } \boxed{2} \text{ for } q.
\]
Quick Answer
The values are \( p = 2 \) and \( q = 2 \).
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