Question

If $f(x)=\frac{1}{2x-4}$ then the point(s) of discontinuity of $f(f(x))$ is/are

• A. $2,\frac{9}{4}$
• B. $3,4$
• C. $1,3$
• D. $4$
• E. $3$

Hint:

Logic Tip: Many students forget to check the continuity of the inner function itself when working with composite functions. Always verify the domain of the innermost expression first before analyzing the larger composition.

Answer 1

The Correct Option is A

Concept:
A composite function $f(g(x))$ is discontinuous at any point where: 1. The inner function $g(x)$ is discontinuous. 2. The outer function $f(u)$ is discontinuous when evaluated at the inner function's output (i.e., $u = g(x)$ is a point of discontinuity for $f$). Here, both the inner and outer functions are $f(x)$.
Step 1: Find the discontinuities of the inner function f(x).
A rational function is discontinuous where its denominator equals zero. Set the denominator of $f(x)$ to zero: $$2x - 4 = 0$$ $$2x = 4$$ $$x = 2$$ So, the first point of discontinuity is $x = 2$.
Step 2: Find the conditions for the outer function's discontinuity.
The composite function is $f(f(x))$. Just like in Step 1, the outer function is undefined when its input equals 2. Therefore, $f(f(x))$ will be discontinuous when the inner function $f(x) = 2$.
Step 3: Solve for x when f(x) = 2.
Set the original function equal to 2: $$\frac{1}{2x - 4} = 2$$ Cross-multiply to solve for $x$: $$1 = 2(2x - 4)$$ $$1 = 4x - 8$$ Add 8 to both sides: $$9 = 4x$$ $$x = \frac{9}{4}$$
Step 4: Combine all points of discontinuity.
The composite function is discontinuous at the points found in Step 1 and Step 3. The points are $2$ and $\frac{9}{4}$.