Travelling wave and phase velocity

(b) is not a travelling wave because it is not of the form $$f(x-vt)+g(x+vt)$$ It does not satisfy the wave equation.

(d) is correct.

First, I'll try a definition of a wave. In one dimension (e.g., with coordinate z), any solution of (a system of) a linear partial differential equation of the form $$psi(z, t)=psi_0 exp (ikz-iomega t) tag 1$$ where $psi_0$ is the (complex) amplitude, $k$ is the wave vector, and $omega$ is the angular frequency, can be considered to be a "wave" with phase velocity $$v_{ph}=frac {omega }{k}$$ and group velocity $$v_{gr}=frac{partial omega}{partial k}$$ Any linear superposition of such sinusoid waves is again a wave. The functional relation between the frequency $omega$ and the wave vector $k$, $$D(omega, k)=0$$ is called the dispersion relation, where $omega $ and $k$ can be complex. It follows from the partial differential equations defining the wave propagation. Solutions (1) of the proper wave equation have the simple linear dispersion relation $$omega =v_{ph} k$$ with a constant phase velocity, which is equal to the group velocity $$v_{ph}=v_{gr}$$ More complex systems, like a medium with damping, have more complicated (non-linear) dispersion relations so that both phase and group velocities can become frequency (or wave length) dependent. Usually, solutions as eq. (1) are considered propagting wave in frequency and wave vector ranges where both $omega$ and $k$ have real parts, so that real phase and group velocities ($v_{gr} neq 0$) can be defined, which usually signifies the propagation of a signal/energy.

The function (b) of the question seems not to be expressible as a linear combination of $t$ an $z$. It thus cannot be regarded to represent a propagating wave. Function (d) should be a propagating wave.

Anybody who thinks that this is a still wanting description of a traveling wave is asked to improve it or provide a better definition.