Similar Functions

Similar solution as @Ephram Chun 's

Let $g(x) = ax^\blue{2a+1} + bx^\blue{2b+1} + cx^\blue{2c+1} + dx^\blue{2d+1}$ . Note that $g(x)$ is an odd function . That is $g(-x) = - g(x)$ . Then we have:

$\begin{aligned} f(x) & = g(x) + 61 \\ f(13) & = g(13) + 61 = -13 \\ \implies g(13) & = f(13)-61 = - 13 -61 = - 74 \\ f(-13) & = \blue{g(-13)} + 61 & \small \blue{\text{Since }g(x) \text{ is odd,}} \\ & = \blue{-g(13)} + 61 \\ & = \blue{-(-74)} + 61 \\ & = \boxed{135} \end{aligned}$

We can write out $f(13)$ as $-13=a(13)^{2a+1}+b(13)^{2b+1}+c(13)^{2c+1}+d(13)^{2d+1}+61$ which is equal to $-74=a(13)^{2a+1}+b(13)^{2b+1}+c(13)^{2c+1}+d(13)^{2d+1}$ . Let's write the form out for $f(-13). f(-13)=a(-13)^{2a+1}+b(-13)^{2b+1}+c(-13)^{2c+1}+d(-13)^{2d+1}+61$ we can subtract $61$ from both sides to get $f(-13)-61=a(-13)^{2a+1}+b(-13)^{2b+1}+c(-13)^{2c+1}+d(-13)^{2d+1}$ We see that $a(-13)^{2a+1}+b(-13)^{2b+1}+c(-13)^{2c+1}+d(-13)^{2d+1}$ is the opposite of $a(13)^{2a+1}+b(13)^{2b+1}+c(13)^{2c+1}+d(13)^{2d+1}$ Therefore the answer is $f(-13)=-74*-1+61=\boxed{135}$