Question

For which of the following functions g is g(z) = g(1-z) for all z?

• A. g(z) = 1-z
• B. g(z) = 1-z²
• C. g(z) = z² - (1-z)²
• D. g(z) = z²(1-z)²
• E. g(z) = z / (1-z)

Hint:

The expression \( (1-z) \) is a transformation that reflects the function across the line \( z=1/2 \). When checking symmetry like \( g(z) = g(1-z) \), look for expressions where \( z \) and \( (1-z) \) can be interchanged without changing the function's value. Products like \( z(1-z) \) or sums like \( z + (1-z) \) are good candidates. In option (D), both \(z^2\) and \((1-z)^2\) are present, and swapping them does not change their product.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The condition \( g(z) = g(1-z) \) means the function is symmetric about the vertical line \( z = 1/2 \). To find the correct function, we need to substitute \( (1-z) \) in place of \( z \) into each of the given functions and check if the resulting expression simplifies back to the original function \( g(z) \).
Step 2: Key Formula or Approach:
For each option, calculate \( g(1-z) \) and compare it to \( g(z) \).
Step 3: Detailed Explanation:
Let's test each option:
(A) g(z) = 1-z
\( g(1-z) = 1 - (1-z) = 1 - 1 + z = z \).
Since \( z \neq 1-z \) for all z, this option is incorrect.
(B) g(z) = 1-z²
\( g(1-z) = 1 - (1-z)^2 = 1 - (1 - 2z + z^2) = 1 - 1 + 2z - z^2 = 2z - z^2 \).
Since \( 2z - z^2 \neq 1 - z^2 \) for all z, this option is incorrect.
(C) g(z) = z² - (1-z)²
\( g(1-z) = (1-z)^2 - (1-(1-z))^2 = (1-z)^2 - (z)^2 = - (z^2 - (1-z)^2) = -g(z) \).
Since \( g(1-z) = -g(z) \), not \( g(z) \), this option is incorrect.
(D) g(z) = z²(1-z)²
\( g(1-z) = (1-z)^2 \times (1 - (1-z))^2 \).
\( g(1-z) = (1-z)^2 \times (z)^2 \).
Since multiplication is commutative, \( (1-z)^2 z^2 = z^2 (1-z)^2 \), which is the original function \( g(z) \).
This option satisfies the condition \( g(z) = g(1-z) \).
(E) g(z) = z / (1-z)
\( g(1-z) = \frac{1-z}{1-(1-z)} = \frac{1-z}{1-1+z} = \frac{1-z}{z} \).
Since \( \frac{1-z}{z} \neq \frac{z}{1-z} \) for all z, this option is incorrect.
Step 4: Final Answer:
The correct function is g(z) = z²(1-z)².