Question

Suppose that f is a one-to-one function, and f^-1 is its inverse. Suppose also that h(x)=4 and g(x)=x^2 + xsecx. then which of the following do we NOT know to be true?

I'll post the MC options below.

Answer 1

A:
Using the property
\[f(f^{-1}("anything")) = "anything"\]
\[\rightarrow f(f^{-1}(h(x)) = h(x) = 4\]
Therefore A is True

B:
We know that
\[h(x) = h("anything") = 4\]
Output is 4 regardless of input
Therefore
\[h(g(g(x))) = 4\]
B is True

C:
\[g(h(f^{-1}(x))) = g(h("anything")) = g(4)\]
\[g(4) = 4^2 + 4 \sec(4)\]
Therefore C is True


D:
Here you have a product of 2 functions
\[h(x)*f^{-1}[ f^{-1} [ f(f(g(x)))]]\]
Lets rewrite it subbing the innermost f(g(x) with "anything"
\[\rightarrow 4*f^{-1} [f^{-1}(f("anything"))]\]
Using inverse function property
\[4*f^{-1}["anything"] = 4*f^{-1}(f(g(x))\]
Reappling property again
\[\rightarrow 4*g(x) = 4x^2 +4x \sec(x)\]

Therefore D is True

E:
\[f[g(f^{-1}(x))] = ?\]
This cannot be simplified since f and its inverse are disconnected with "g" in between them.
Therefore the property does not hold for this expression and you cannot assume it equals g(x)

E is FALSE