You need to remember that is a function strictly increases, then the function is injective. Since `f(x)=x+3^x` is a sum of increasing functions, then the function f(x) is injective.
You need to remember that you may prove that a function is bijection if a parallel line to x axis intercepts the graph of function just one single time.
Hence you need to sketch the graph of function `f(x)=x+3^x` such that:
Sketching parallel lines to x axis, y=-5, y=-4.5, y=-4, ..., y=1, ..., y=10,.. you notice that these parallel lines intercept the graph one time each, hence, the function is continuous.
Hence, since the function is injective and surjective, then f(x) is a bijection.