Q.Find the domain for which the functions $f(x)=2x^2-1$ and $g(x)=1-3x$ are equal.

Solve $f(x)=g(x)$: $2x^2-1=1-3x\Rightarrow 2x^2+3x-2=0\Rightarrow x=\dfrac{1}{2},\,-2$.

Two functions $f$ and $g$ are equal exactly at those values of $x$ where $f(x)=g(x)$.

Step 1 — Set equal.

$2x^2-1=1-3x.$

Step 2 — Rearrange to a quadratic.

$2x^2+3x-1-1=0\ \Rightarrow\ 2x^2+3x-2=0.$

Step 3 — Factorise.

$2x^2+4x-x-2=0\ \Rightarrow\ 2x(x+2)-1(x+2)=0\ \Rightarrow\ (x+2)(2x-1)=0.$

Step 4 — Solve.

$x+2=0\Rightarrow x=-2,\qquad 2x-1=0\Rightarrow x=\frac{1}{2}.$

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