Question

Show that $f : \mathbf{R}_+ \to [-5, \infty)$ given by $f(x) = 4x^2 + 4x - 5$ is both one-one and onto where $\mathbf{R}_+ = [0, \infty)$. Also, find $p \in \mathbf{R}_+$ such that $f(p) = 3$.

Hint:

To complete the square easily for quadratics like $4x^2 + 4x - 5$, rewrite it as $(2x+1)^2 - 1 - 5 = (2x+1)^2 - 6$. Setting this equal to $y$ makes isolating $x$ direct and bypasses the full quadratic formula setup entirely.

Answer 1