If g(x) is a decreasing function on R and f(x) = tan–1 {g(x)}. State whether f(x) is increasing or decreasing on R.
g(x) is decreasing on R.
f(x) = tan-1x is an increasing function.
f o g(x) = f(g(x)) = tan-1(g(x))
is a decreasing function
Composite of two functions,
Glossary –
↑ - Increasing
↓ - For decreasing f(x).
x1 < x2
g (x1) > g (x2)
Applying, tan-1 on both the sides, of the mentioned equation,
tan-1 {g (x1)} > tan-1 {g (x2)}
f (x1) > f (x2)
Hence it is decreasing on R.
1