Question

Which of the following statements describe a function and its inverse? Select true (T) or false (F).

a. The compositions f(g(x)) and g(f(x)) will simplify to z if the functions are inverses.
b. If the point (x, y) is on the graph of f(x), the point (y, x) is on the graph of g(x) if the two functions are inverses.
c. A function and its inverse are reflections across the z-axis.
d. If the inverse of the equation f(x) is found, the inverse is g(x) if f(g(x)) = g(f(x)).

Answer 1

The statements describe the relationship between a function and its inverse. For functions f(x) and g(x) to be inverses, f(g(x)) and g(f(x)) must simplify to x, and points (x, y) from f(x) correspond to points (y, x) on g(x). However, they are reflections across the y=x line, not the z-axis.

Step-by-step explanation:

The question pertains to the properties of a function and its inverse function.

• (a) True - If functions f(x) and g(x) are inverses of each other, then f(g(x)) and g(f(x)) should both simplify to x, not z as mentioned. However, given the context, the statement is conceptually true but includes a typo.
• (b) True - If (x, y) is on the graph of f(x), then (y, x) will be on the graph of g(x), assuming f and g are inverse functions.
• (c) False - A function and its inverse are reflections across the y=x line, not the z-axis.
• (d) True - If f(g(x)) = x and g(f(x)) = x, then g(x) is indeed the inverse of f(x).

To correct the error, the correct statement for (c) is: A function and its inverse are reflections across the y=x line.