We expect to see a ___ for the graph of a composition of a function and its inverse function, if the domain of each is all real numbers.
line
A graph of a composition of a function and its inverse function, if the domain of each is all real numbers, should result in a straight line with a slope of 1 passing through the origin (0,0). This is because the composition of a function with its inverse always results in the input value (x) being equal to the output value (y). In other words, if we let f be the function and g be its inverse, then we have:
(f ∘ g)(x) = x and (g ∘ f)(x) = x
When we plot these as points on a graph (x, f ∘ g(x)) and (x, g ∘ f(x)), we see that they lie on a straight line passing through the origin with slope 1. This is because the composition of the function and its inverse undo each other, resulting in a linear relationship between the input and output values. This property is true for all invertible functions with domain and range of real numbers.
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