Detect and write the corresponding elementary functions
a) $$f(x)=e^{2\sin x}$$
b) $$f(x)=\sqrt{\sin(x^2-x+2)}$$
See development and solution
Development:
a) $$g(x)=e^x; \ \ h(x)=2x; \ \ t(x)=\sin(x)$$
The composition is the following one: $$f(x)=g(h(t(x)))$$
b) $$g(x)=\sqrt{x}; \ \ h(x)=\sin(x); \ \ t(x)=x^2-x+2$$
In this case the function $$t(x)$$ is not an elementary function, but it is a sum of elementary functions. How does the composition work?
The composition is the following one: $$f(x)=g(h(t(x)))$$
Solution:
a) $$g(x)=e^x; \ \ h(x)=2x; \ \ t(x)=\sin(x) \Rightarrow f(x)=g(h(t(x)))$$
b) $$g(x)=\sqrt{x}; \ \ h(x)=\sin(x); \ \ t(x)=x^2-x+2 \Rightarrow f(x)=g(h(t(x)))$$