The function f: R $\to$ R defined by f(x) = 2x^2 + $x$ - 1 is ________.

Step 1 (Analyze One-to-One / Injectivity): A function $f$ is one-to-one if $f(x_1)=f(x_2)⟹x_1=x_2$.
For $f(x)=2x^2+x-1$, this is a quadratic function represented by a parabola opening upwards.

• If we find two points with the same output: $f(x)=0⟹2x^2+x-1=0⟹(2x-1)(x+1)=0⟹x=0.5$ and $x=-1$ both give $f(x)=0$.

• Since $f(0.5)=f(-1)=0$ but $0.5\ne -1$, the function is not one-to-one.

Step 2 (Analyze Onto / Surjectivity): A function $f:\mathbb{R}\to \mathbb{R}$ is onto if the range of the function is the entire set of real numbers $\mathbb{R}$.

• For the parabola $f(x)=2x^2+x-1$, the vertex represents the absolute minimum point:

• The range of the function is strictly bounded below: $[-1.125,\infty )$. Since there are no real numbers $x$ that map to values below $-1.125$ (e.g., $f(x)=-2$ has no real solution), the function is not onto.

Step 3 (Conclusion): The function is neither one-to-one nor ontoSurjective.