Classifying functions from their graphs

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For each of the following graphs (functions $\mathbb{R}\to\mathbb{R}$) decide whether it is an injective, surjective, bijective function, or none of these. (graphs in the original test)

• (a) parabola $y=x^2-1$ (vertex at $(0;-1)$, opening upward);
• (b) exponential $y=e^{x/3.1}$ (increasing, always positive);
• (c) $y=3\tanh\!\left(\tfrac{x}{3}\right)$ (increasing, between the asymptotes $y=-3$ and $y=3$);
• (d) line $y=\tfrac{x}{2}+1$.

Solution

Use the horizontal line test (injectivity: every horizontal line meets the graph at most once) and look at the image set (surjectivity onto $\mathbb{R}$: the image is all of $\mathbb{R}$).

• (a) The parabola is symmetric: the same $y$ comes from two $x$ values ⇒ not injective; also $y\ge-1$ ⇒ not surjective. None of these.
• (b) The exponential is strictly increasing ⇒ injective; but $y>0$ ⇒ not surjective. Injective not surjective.
• (c) The hyperbolic tangent is strictly increasing ⇒ injective; its image is $(-3;3)$ ⇒ not surjective. Injective not surjective.
• (d) The (non-constant) line is increasing and takes all real values ⇒ injective and surjective ⇒ bijective. $\boxed{\ (a)\ \text{none};\ \ (b)\ \text{injective};\ \ (c)\ \text{injective};\ \ (d)\ \text{bijective}\ }$