Continuity and Differentiability
Continuity — Finding x from Composite Function
nta_pyq_2026_jan
Grade 12
Question:
Let $f(x) = \begin{cases} \dfrac{ax^2+2ax+3}{4x^2+4x-3} & x \neq -\dfrac{3}{2},\,\dfrac{1}{2} \\ b & x = -\dfrac{3}{2},\,\dfrac{1}{2} \end{cases}$ be continuous at $x=-\dfrac{3}{2}$. If $f\circ f(x)=\dfrac{7}{5}$, then $x$ is equal to:
Step-by-Step Solution
Key Concept: $4x^2+4x-3=(2x+3)(2x-1)$. For continuity at $x=-3/2$: numerator must vanish $\Rightarrow a(9/4)-3a+3=0\Rightarrow a=4$. Then $f(x)=\dfrac{2x+1}{2x-1}$ for $x\neq-3/2$.
$a=4$, $f(x)=\dfrac{2x+1}{2x-1}$. $f\circ f(x)=\dfrac{7}{5}\Rightarrow x=1$.
Correct Answer: 4