Question

If f(x) = xx - 1 and f(g(x)) = 4x - 2x - 1, then:

i. g(x) = 2x - 1
ii. g(x) = 1 - 2x
iii. g(x) = -2x
Option 1: Only i
Option 2: Only ii and iii
Option 3: Only i and iii
Option 4: I, ii, iii
Option 5: None of the above

Answer 1

By substituting the given functions for g(x) into the expression for f(x) and simplifying, it is determined that only g(x) = 2x - 1 results in the expression provided for f(g(x)), so only option i is correct.

Step-by-step explanation:

The student has asked to determine which of the given options for g(x) are correct, given that f(x) = xx - 1 and f(g(x)) = 4x - 2x - 1. To find the correct function g(x), we need to substitute the options into f(x) and see if the result matches f(g(x)).

Let's try the first option i. g(x) = 2x - 1:
f(g(x)) = f(2x - 1) = (2x - 1)(2x - 1) - 1.

The expression for f(g(x)) provided in the question simplifies to 2x - 1 which matches the result of substituting option i, therefore g(x) = 2x - 1 could be a correct function.

Now let's check option ii. g(x) = 1 - 2x:
f(g(x)) = f(1 - 2x) = (1 - 2x)(1 - 2x) - 1.

This does not match the given expression for f(g(x)), so option ii cannot be correct. Finally, let's check option iii. g(x) = -2x:
f(g(x)) = f(-2x) = (-2x)(-2x) - 1.

This also doesn't match the given expression for f(g(x)), so option iii is incorrect as well. So we conclude that only option i is correct and the correct answer is Option 1: Only i.