f(x) = ax^7 + bx^3 + cx -7

f(7) = a(7)^7 + b(7)^3 + c(7) -7 = 7

==> (7^7)a + (7^3)b + 7c = 14.........(1)

f(-7) = a(-7)^7 + b (-7)^3 + c(-7)-7

= -1(a(7^7) - b (7^3) - c(7) - 7

= -[ a(7^7) + b(7^3) + 7c)] -7

But from (1) we know that inside brackets equals 14

= => -(14) -7 = -21

==> **f(-7) = -21**

f(7) = 7

Therfore f(7) = a(7)^7+b(7)^3+c(7) -7 = 7...........(1)

f(-7) = a(-7)^7+b(-7)^7+c(-7) -7

= -{a(7)^7+b(7)^3+c(7)}-7

= -{a(7)^7+b(7)^3+c(7)-7} -14

= -{7} - 14 = -21.

If f(x)=a*x^7 + b*x^3 + c*x – 7, then

f(-x)= a*(-x)^7 + b*(-x)^3 + c*(-x) –7

f(-x)=-a*x^7 - b*x^3 - c*x –7

If we calculate the sum

f(x)+ f(-x)= a*x^7 + b*x^3 + c*x – 7-a*x^7 - b*x^3 - c*x –7

f(x)+ f(-x)=-14

From the sum above, we’ve noticed that regardless the value of x, the sum will have always the same value “-14”.

So the sum is not depending on x. The conclusion would b that:

f(7)+ f(-7)=-14

But f(7)=7, the value being given in the enunciation. So we’ll substitute it in the sum:

7+ f(-7)=-14

f(-7)=-14-7

**f(-7)=-21**