Functions
Equation f(x) = f(g(x)) — counting roots and their sum
MJAT_TS6_P1
Grade 12
Question:
Let $f(x)=x^2-12x+3$ defined on $[-5,5]$. Let $m$ and $n$ be the number of roots and sum of roots respectively of $f(x)=f\!\left(\dfrac{x+8}{x-1}\right)$. Then $m+n$ equals:
Step-by-Step Solution
Key Concept: $f(x)=f(g)$ where $g=\frac{x+8}{x-1}$ means either $x=g$ or $x+g=12$ (using $f(x)=f(y)\Leftrightarrow x=y$ or $x+y=12$ for this quadratic). Case 1: $x=\frac{x+8}{x-1}\Rightarrow x(x-1)=x+8\Rightarrow x^2-2x-8=0\Rightarrow x=4,-2$. Case 2: $x+\frac{x+8}{x-1}=12$.
$m+n=\mathbf{4}$.
Correct Answer: 4