Functions are not always a straight line, no. a parabola is a function, and it forms a "U" shape. a straight line is called a LINEAR function, but there are many non-linear functions.
y = x does represent a function. while it doesn't look like it, this equation is still in slope-intercept, y = mx + b, form. the slope is just 1 and the y-intercept is 0, so what's left is the equation y = x. the definition of a function requires an output for every input, and this equation still works. try plugging in 1 for x:
y = (1) y = 1
so when your input is 1, your output is also 1. look at the graph of y = x and you'll understand why this works. furthermore, while looking at the graph, you can double check that this is a function with the vertical line test (which makes sure that x values aren't repeated).
