Problem Solving

teejaycrown wrote:For which of the follwoing functions f is f(x) = f (1 - x) for all x?

a. F(x) = 1 -x
b. f(x) = 1 - x ^2
c. f(x) = x^2 - (1-x)^2
d. f(x) = x^2 (1-x^2)
e. f(x) = x/1-x

Let us look at each of the options:

(A) f(x) = 1-x
f(1-x) = 1 - (1-x) = x; FALSE

(B) f(x) = 1-x^2
f(1-x) = 1 - (1-x)^2 = 1 - (1 - 2x + x^2) = 2x - x^2; FALSE

(C) f(x) = x^2 - (1-x)^2
f(1-x) = (1-x)^2 - (1 - (1-x))^2 = 1 - 2x + x^2 - (x)^2 = 1 - 2x; FALSE

(D) f(x) = x^2 * (1-x)^2
f(1-x) = (1 - x)^2 * (1 - (1 - x))^2 = (1 - x)^2 * (x)^2; TRUE

(E) f(x) = x/(1-x)
so f(1-x) = (1-x)/(1-(1-x)) = (1-x)/x; FALSE

The correct answer is D.

teejaycrown wrote:For which of the follwoing functions f is f(x) = f (1 - x) for all x?

a. F(x) = 1 -x
b. f(x) = 1 - x ^2
c. f(x) = x^2 - (1-x)^2
d. f(x) = x^2 (1-x^2)
e. f(x) = x/1-x

Picking Numbers Approach:

If x = 1, then 1 - x = 1 - 1 = 0

We have to find for which of the functions f(1) = f(0).

(A) f(1) = 1 - 1 = 0, and f(0) = 1 - 0 = 1; NOT True

(B) f(1) = 1 - 1² = 0, and f(0) = 1 - (0)² = 1; NOT True

(C) f(1) = 1² - (1-1)² = 1, and f(0) = 0² - (1 - 0)² = -1; NOT True

(D) f(1) = 1 * (1 - 1²) = 0, and f(0) = 0; TRUE

The correct answer is D.