Let f(x)=px+q, where p and q are real numbers. Find p+q if f(f(f(x)))=8x+21. Thank you.
To find p + q, we will solve the equation f(f(f(x))) = 8x + 21 using the given function f(x) = px + q.
Step 1: Evaluate f(x) for the given input
Start by substituting f(x) into the equation f(f(f(x))) = 8x + 21:
f(f(f(x))) = 8x + 21 becomes f(px + q) = 8x + 21
Step 2: Simplify the expression
To simplify, we will substitute px + q into the function f(x):
f(px + q) = p(px + q) + q
Simplifying further:
= p^2x + pq + q
Step 3: Evaluate f(px + q)
Now, substitute f(px + q) back into the original equation f(f(f(x))) = 8x + 21:
p^2(px + q) + pq + q = 8x + 21
Step 4: Simplify the equation
Distribute p^2 through px + q:
p^3x + p^2q + pq + q = 8x + 21
Step 5: Combine like terms
Combine all the terms with x and all the constant terms:
(p^3 - 8)x + (p^2 + p + 1)q = 21
For the equation to hold for all values of x, the coefficients of x and q must be equal to zero.
Step 6: Set up two equations
Set p^3 - 8 = 0 and p^2 + p + 1 = 0 to solve for p.
p^3 - 8 = 0 can be rewritten as p^3 = 8, which means p = 2.
Step 7: Solve for p + q
Now that we know p = 2, substitute it into the equation p^2 + p + 1 = 0:
(2)^2 + 2 + 1 = 4 + 2 + 1 = 7
Therefore, p + q = 2 + q = 7.
So, q = 7 - 2 = 5.
Thus, p + q = 2 + 5 = 7.
Therefore, p + q equals 7.