Functions
Solving reciprocal functional equation
nta_pyq_2025_apr
Grade 12
Question:
Let f$(x) + 2f$( 1$) = x$$2 + 5$and$2g(x) - 3g$( 1$) = x$,$x > 0$. If$\alpha$= $\int$ 2 f (x)dx , and$\beta$= $\int$ 2 g(x)dx , then the x 2 1 1 value of 9$\alpha$+$\beta$is:
Step-by-Step Solution
Key Concept: Write the given equation with the transformed input and solve the resulting simultaneous equations for the function value.
f$(x) + 2f$( 1$) = x$$2 + 5$(4) x 1 1 f ($) + 2f$(x) = + 5 x x 2 2 2 x 5 f (x) = - + 2 3x 3 3 2 2 2 x 5$\alpha$= $\int$ ( - + ) dx 2 3x 3 3 1 2 3 2 x 5x (- - + ) 3x 9 3 1 1 8 10 2 1 5 - - + + + - 3 9 3 3 9 3 7 11$\alpha$= 2 - = 9 9 1$2g(x) - 3g$($) = x$2 1 1 g( ) = - 2 2 x 3 g(x) = - 2 4 2 x 3$\beta$= $\int$$( - )$dx 2 4 1 2 2 x 3x 3 1 3$( - ) = 1$- - + = 0 4 4 2 4 4 1 9$\alpha$+$\beta$= 11 option (4)
Correct Answer: 4