2580.41 – A Seventh Power Equation

Let f(x) = ax7 + bx3 + cx −5f(x) = ax^7 + bx^3 + cx - 5, where a, b,a, b, and cc are constants.

If f(−7)=7f(-7) = 7, then what is f(7)f(7)?

  1. -17
  2. -7
  3. 14
  4. 21
  5. not uniquely determined
Solution

Since f(x) = ax7 + bx3 + cx –5,f(x) = ax^7 + bx^3 + cx – 5, and f(−7)=7f(-7) = 7 then

f(−7)= a(−7)7 + b(−7)3 + c(−7)–5,7= a(−7)7 + b(−7)3 + c(−7)−5,7=−a77 − b73 − c(7)−57=−(a77−b73−c(7)−5)−107=−f(7)−10.\begin{aligned} f(-7) &=& a(-7)^7 + b(-7)^3 + c(-7) – 5, \\ 7 &=& a(-7)^7 + b(-7)^3 + c(-7) - 5, \\ 7 &=& -a7^7 - b7^3 - c(7)- 5 \\ 7 &=& -(a7^7 - b7^3 - c(7)- 5) - 10 \\ 7 &=& -f(7) - 10. \end{aligned}

So f(7)=−10−7=−17.f(7) = -10 - 7 = -17. The answer is (a).