1. What happens if we graph both f and f^{-1} on the same set of axes, using the x-axis for the input to both f and f^{-1}? [Suggestion: go to www.desmos.com/calculator and type y=x^3 {-2 < x < 2}, y=x^{1/3} {–2 < x < 2}, and y = x {–2 < x < 2}, and describe the relationship between the three curves.]
f(x)=x^3 and f^-1(x)= ∛x are inverse functions, and can also be expressed as f^-1=x^1/3. If these two functions are placed on the same set of axes, they are reflections of each other, and the line y = x is the axis of symmetry. Also, all three functions share the same final behavior, so it is not easy to graph both f and f^{-1} on the same set of axes.
Picture 1 [Created on www.desmos.com/calculator]
2. Then post your own example discussing the difficulty of graph both f and f^{-1} on the same set of axes.
f(x)=x^4 and f^-1(x)= ∜-1x are inverse functions, and can also be expressed as f^-1=x^1/4. If these two functions are placed on the same set of axes, they are reflections of each other, and the line y = x is the axis of symmetry. Also, all three functions share the same final behavior, so it is not easy to graph both f and f^{-1} on the same set of axes.
Picture 2 [Created on www.desmos.com/calculator]
Suppose f:R \rightarrow R is a function from the set of real numbers to the same set with f(x)=x+1. We write f^{2} to represent f \circ f and f^{n+1}=f^n \circ f.
3. Is it true that f^2 \circ f = f \circ f^2? Why?
For example, f1 = x+1
f2 = (x+1)+1 = x+2
f2 \circ f = (x+1)+2 = x+3
inverse
f \circ f2 = (x+2)+1 = x+3
Therefore, it is TRUE that f^2 \circ f = f \circ f^2.
4. Is the set {g:R \rightarrow R l g \circ f=f \circ g} infinite? Why?
g \circ f(x) = g(f(x)) ⇒ f \circ g(x) = f(g(x))
The set is INFINITE because changing any of the arguments does not change the final result, and they are inverses of each other.
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