34. The function f is defined by f(c) = 1/(x - 2) - 1/x where x ≠ 2 and x ≠ 0. If f(c) = 1/12, which of the following could be the value of c?. Indicate all such values.
Answer: C,E
C and E could be the values of c.
When solving for c in the equation f(c) = 1/(x - 2) - 1/x = 1/12, we find that both c = -4 and c = 6 satisfy this equation.
A) -12
Substituting c = -12 into the function results in f(-12) = 1/(-12 - 2) - 1/(-12), which simplifies to 1/(-14) + 1/12. This does not equal 1/12, therefore -12 is not a valid solution.
B) -6
For c = -6, f(-6) = 1/(-6 - 2) - 1/(-6), which yields 1/(-8) + 1/6. This does not satisfy the equation f(c) = 1/12, making -6 an incorrect choice.
C) -4
When c = -4, we find f(-4) = 1/(-4 - 2) - 1/(-4) = 1/(-6) + 1/4. This does simplify to 1/12, confirming that -4 is indeed a valid solution.
D) 4
Substituting c = 4 into the function gives f(4) = 1/(4 - 2) - 1/4 = 1/2 - 1/4, which equals 1/4. This does not match the required value of 1/12, thus 4 is not a valid option.
E) 6
For c = 6, f(6) = 1/(6 - 2) - 1/6 = 1/4 - 1/6. This computes to 1/12, confirming that 6 is a valid solution.
Conclusion
C and E are the only values that satisfy the function equation f(c) = 1/12 when substituted into the function. All other options fail to meet this criterion, making them invalid solutions. Thus, only -4 and 6 can be considered as correct answers.