The functions f and g are defined by f : x → 2x + ln 2, g : x → e^(2x). Find the composite function gf, sketch its graph and find its range.

Composition of functions is when one function is inside of another function. That is, we replace each occurrence of x found in the outside function with the inside function. In our example, we start by writing : gf(x) = e^(2(2x + ln 2)) . We expand the brackets on the exponent and we split the function in two exponentials : gf(x)=e^(4x)e^(2ln(2)) . Now we use 2 properties of the logarithmic functions, in order to simplify our function. The first property we use is that when we have ln(x^n), we can take the power outside of the natural logarithm to get nln(x). The second property is that ln(e) is always equal to 1. Thus, by applying the above properties in our example we get : gf(x) = 4e^(4x) . Next, we need to sketch the graph of the gf(x). The sketch is nothing but the usual graph of the exponential function e^x, but in our case the point of intersection with the y-axis will be equal to 4, since when we set x=0 we get y=4. Finally, the range of the function is the following : gf(x)>0 . This is because the graph is asymptotic to the x-axis as x approaches negative infinity and it increases without bound as x approaches positive infinity. Thus we have successfully answered all parts of the question.