Question:
If $f(x) = \frac{x+2}{3x-1}$, then $f(f(x))$ is:
If $f(x) = \frac{x+2}{3x-1}$, then $f(f(x))$ is:
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If \( f(f(x)) = x \), then the function is its own inverse. This often happens with linear fractional transformations where the sum of the main diagonal elements (in this case $1$ and $-1$) is zero.
Updated On: May 6, 2026
- \( x \)
- \( -x \)
- \( \frac{1}{x} \)
- \( -\frac{1}{x} \)
- \( 0 \)
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The Correct Option is A
Solution and Explanation
Concept:
Composite functions involve substituting one function into another. To find \( f(f(x)) \), we replace every instance of \( x \) in the expression for \( f(x) \) with the entire expression of \( f(x) \) itself.
Step 1: Set up the composite expression.
\[ f(f(x)) = \frac{f(x) + 2}{3f(x) - 1} \] Substitute \( f(x) = \frac{x+2}{3x-1} \): \[ f(f(x)) = \frac{\frac{x+2}{3x-1} + 2}{3\left(\frac{x+2}{3x-1}\right) - 1} \]
Step 2: Simplify the numerator and denominator.
Numerator: \[ \frac{x+2 + 2(3x-1)}{3x-1} = \frac{x+2+6x-2}{3x-1} = \frac{7x}{3x-1} \] Denominator: \[ \frac{3(x+2) - 1(3x-1)}{3x-1} = \frac{3x+6-3x+1}{3x-1} = \frac{7}{3x-1} \]
Step 3: Combine and solve.
\[ f(f(x)) = \frac{\frac{7x}{3x-1}}{\frac{7}{3x-1}} = \frac{7x}{7} = x \]
Step 1: Set up the composite expression.
\[ f(f(x)) = \frac{f(x) + 2}{3f(x) - 1} \] Substitute \( f(x) = \frac{x+2}{3x-1} \): \[ f(f(x)) = \frac{\frac{x+2}{3x-1} + 2}{3\left(\frac{x+2}{3x-1}\right) - 1} \]
Step 2: Simplify the numerator and denominator.
Numerator: \[ \frac{x+2 + 2(3x-1)}{3x-1} = \frac{x+2+6x-2}{3x-1} = \frac{7x}{3x-1} \] Denominator: \[ \frac{3(x+2) - 1(3x-1)}{3x-1} = \frac{3x+6-3x+1}{3x-1} = \frac{7}{3x-1} \]
Step 3: Combine and solve.
\[ f(f(x)) = \frac{\frac{7x}{3x-1}}{\frac{7}{3x-1}} = \frac{7x}{7} = x \]
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