Question

If $f(x)=\frac{3x-2}{5x+3}$ where $f:\mathbb{R}-\{-\frac{3}{5}\}\rightarrow\mathbb{R}$ is defined, then $f\circ f(1)$ is

• A. 1
• B. $\frac{-13}{29}$
• C. $\frac{13}{29}$
• D. -1

Hint:

Logic Tip: Alternatively, finding the composite function $f(f(x))$ first yields $\frac{3(\frac{3x-2}{5x+3})-2}{5(\frac{3x-2}{5x+3})+3} = \frac{-x-12}{30x-1}$. Substituting $x=1$ directly gives $\frac{-1-12}{30-1} = \frac{-13}{29}$. Both methods are robust, but sequential numerical substitution usually poses fewer chances for algebraic sign errors.

Answer 1

The Correct Option is B

Concept:
The composition of a function with itself at a specific value, denoted $f \circ f(x)$, evaluates the function using the result of the first evaluation as the input for the second evaluation. $f \circ f(1) = f(f(1))$. We can either find the general expression for $f(f(x))$ or evaluate sequentially. Let's evaluate sequentially.
Step 1: Evaluate the inner function, f(1).
Given $f(x) = \frac{3x - 2}{5x + 3}$. Substitute $x = 1$: $$f(1) = \frac{3(1) - 2}{5(1) + 3}$$ $$f(1) = \frac{3 - 2}{5 + 3} = \frac{1}{8}$$
Step 2: Evaluate the outer function, f(f(1)).
Now we must find $f(f(1))$, which is $f\left(\frac{1}{8}\right)$. Substitute $x = \frac{1}{8}$ back into the original function definition: $$f\left(\frac{1}{8}\right) = \frac{3(\frac{1}{8}) - 2}{5(\frac{1}{8}) + 3}$$ $$= \frac{\frac{3}{8} - 2}{\frac{5}{8} + 3}$$ Find common denominators for the numerator and denominator fractions: $$= \frac{\frac{3 - 16}{8{\frac{5 + 24}{8$$ $$= \frac{-\frac{13}{8{\frac{29}{8$$ Cancel out the common denominator of 8: $$f(f(1)) = -\frac{13}{29}$$