Question

Check whether the function \[ f:\mathbb{R}\setminus\{3\}\rightarrow\mathbb{R} \] defined by \[ f(x)=\frac{x-2}{x-3} \] is onto (surjective) or not.

Hint:

For any rational function of the form \( f(x) = \frac{ax+b}{cx+d} \), the horizontal asymptote is always given by \( y = \frac{a}{c} \). The function will never be able to attain this specific value, meaning its range will automatically exclude \( \frac{a}{c} \). Here \( a=1, c=1 \), so \( y = 1 \) is excluded.

Answer 1