Functions
Additive and multiplicative functions — combined expression
MJAT_TS7_P2
Grade 12
Question:
Let $f:\mathbb{R}\to\mathbb{R}$ with $f(x+y)=f(x)+f(y)$ for all $x,y$, and $g:\mathbb{R}\to(0,\infty)$ with $g(x+y)=g(x)g(y)$ for all $x,y$. If $f(-5/3)=12$ and $g(-1/3)=2$, then $\dfrac{f(1/4)+g(-2)-8}{g(0)}$ equals:
Step-by-Step Solution
Key Concept: $f$ is additive: $f(x)=cx$. $f(-5/3)=c(-5/3)=12\Rightarrow c=-36/5$. $f(1/4)=-36/(5\cdot 4)=-9/5$. $g$ is multiplicative: $g(x)=a^x$. $g(-1/3)=a^{-1/3}=2\Rightarrow a^{1/3}=1/2\Rightarrow a=1/8$. $g(-2)=a^{-2}=64$. $g(0)=1$.
Answer: $\mathbf{51}$.
Correct Answer: 51