Visualisation of composite real functions using 3D
This applet helps visualise a composite function f(g(x)) and its implied domain and range.
Initially we see the 'inner' function g(x) plotted vs x, shown in green. What is its domain and range?
Cliek 'View f(z) vs z' to switch to a plot of the 'outer' function f(z) (blue) plotted against a new axis z. What are its domain and range?
To find the implied domain and range of f(g(x)), we look at the intersection of the range of g and the domain of f. If you look closely, you should be able to see the range of g somewhere in this picture. (Toggle the 'Show g(x)' checkbox if you need a hint.) Tr to identify what is the intersection of range(g) and domain(f).
The range of f(g(x)) is the range of f when restricted to the intersection of range(g) and domain(f). Can you identify this on the graph? (Toggle the checkbox 'Show f(g(x))' if you need a hint.) Click 'View f(g(x)) vs x' to see the graph of the composite function. What is its range - did you get it correct?
Now let's find the implied domain. Backtrack by clicking 'View f(z) vs z' and then 'View g(x) vs x'. To find the implied domain, we need to restrict the domain of g so that its range lies within the domain of f.
Can you see the domain of f somewhere in this picture? (Click'Show f(z)' or 'View f(z) vs z' if you need a hint.) What is the implied domain of f(g(x))? Click 'View f(g(x)) vs x' to check.