Question

Consider these functions:

f(x) = 2x - 1
g(x) = -1/2x + 1/2
Which statements, if any, are true about these functions?
I. The function f(g(x)) = x for all real x.
II. The function g(f(x)) = x for all real x.
III. Functions f and g are inverse functions.

A. I only
B. II only
C. I, II, and III
D. None of the statements are true

Answer 1

None of the given statements about the functions being inverses of each other are true, as the compositions f(g(x)) and g(f(x)) do not yield the identity function x.

Step-by-step explanation:

To address whether the statements about the functions f(x) = 2x - 1 and g(x) = -1/2x + 1/2 are true, we need to consider if they are inverse functions of each other. For two functions to be inverses, the composition of the functions in either order should yield the identity function, meaning f(g(x)) = x and g(f(x)) = x.

We can evaluate f(g(x)):

• f(g(x)) = f(-1/2x + 1/2) = 2(-1/2x + 1/2) - 1 = -x + 1 - 1 = -x.

This indicates that statement I is not true. Now let's check g(f(x)):

• g(f(x)) = g(2x - 1) = -1/2(2x - 1) + 1/2 = -x + 1/2 + 1/2 = -x + 1.

Therefore, statement II is also not accurate, and consequently, statement III about being inverse functions must be false too.

The correct choice is D. None of the statements are true.