Some general ideas:

Consider the function and the arithmetic that goes into it. Many functions can be directly computed: addition and subtraction, multiplication by a constant.

Multiplying by a joint variable is harder, but usually done by taking advantage of logarithms and exponents.

The inverse of any one-to-one function can be computed by placing it in the feedback path of an amplifier; in effect, the amplifier solves for the input corresponding to the given output.

Time is very valuable. Consider differential equations that might produce the desired function, and how to solve them over time. Use resistors and capacitors to your advantage; integration and differentiation.

You might have quite a lot of trouble calculating sin(x) from first principles, but using an integrator to generate a ramp, a comparator to pick a point on that ramp, and a sampler to pick a point on a synchronized sine wave oscillator, you can easily find this value: given the limitations of time delay, sampling (rather than being a continuous-time function) and oscillator distortion, to name a few, of course.

You can sample points or generate events from these waveforms, which can have further combinational effects: the pulses can be multiplied by another value, to get pulse width multiplication effects. You're limited only by your imagination!

The space of possibilities is much the same as used in the olden days of video games, for generating interesting patterns and movements (especially of animations and enemy positions): creatures might bob up and down or zig-zag across the screen, following simple differential equations that were easily computed (using conditionals, integrals and addition, avoiding expensive operations like multiplication, division or exponentiation). In old games, often particles would only travel in straight (vertical or horizontal) directions; you could always tell the good ones, where they went to the effort to make funny angles, or tracking projectiles, and stuff like that. Only a matter of coming up with the right solution, and packing it in within the required time frame.

Tim