Q.The domain for which the functions defined by $f(x) = 3x^2 - 1$ and $g(x) = 3 + x$ are equal is
(A) $\left\{-1,\ \dfrac{4}{3}\right\}$
(B) $\left\{-1,\ \dfrac{4}{3}\right\}$
(C) $\left(-1,\ \dfrac{4}{3}\right)$
(D) $\left[-1,\ \dfrac{4}{3}\right)$

Concept: Piecewise Function Definition — two functions are equal only when their outputs match for the same input.

We set $f(x) = g(x)$:

Bring all terms to one side:

Factor the quadratic:

So $x = -1$ or $x = \frac{4}{3}$. …

The idea is simple: two functions $f$ and $g$ are said to be equal on a domain if for every $x$ in that domain, $f(x) = g(x)$. Here, both $f$ and $g$ are defined for all real numbers (they are polynomials), so the question reduces to: for which $x$ do their outputs coincide? That means solving the equation $f(x) = g(x)$.

1. Set the functions equal. We have $f(x) = 3x^2 - 1$ and $g(x) = 3 + x$. So:

2. Rearrange into a standard quadratic. Bring all terms to one side:

Simplify:

3. Solve the quadratic. Factor or use the quadratic formula. Here, factoring works nicely:

So:

Giving:

4. Interpret the result. These are the only two real numbers where $f(x) = g(x)$. The domain on which the functions are equal is therefore the set containing exactly these two points. …