Question

If \( f(x) = \frac{3x+4}{5x-7} \), \( x \neq \frac{7}{5} \), and \( g(x) = \frac{7x+4}{5x-3} \), \( x \neq \frac{3}{5} \), then \[ (g \circ f)(3) = \]

• A. \( -3 \)
• B. \( -\frac{1}{3} \)
• C. 3
• D. \( \frac{1}{3} \)

Hint:

When solving function compositions, compute the inner function first and then substitute the result into the outer function.

Answer 1

The Correct Option is C

Step 1: Composition of functions.
To find \( (g \circ f)(3) \), we first compute \( f(3) \) and then substitute the result into \( g(x) \). Step 2: Compute \( f(3) \).
Substitute \( x = 3 \) into \( f(x) \): \[ f(3) = \frac{3(3) + 4}{5(3) - 7} = \frac{9 + 4}{15 - 7} = \frac{13}{8} \] Step 3: Compute \( g(f(3)) = g\left(\frac{13}{8}\right) \).
Substitute \( x = \frac{13}{8} \) into \( g(x) \): \[ g\left(\frac{13}{8}\right) = \frac{7\left(\frac{13}{8}\right) + 4}{5\left(\frac{13}{8}\right) - 3} = \frac{\frac{91}{8} + 4}{\frac{65}{8} - 3} \] Simplifying: \[ g\left(\frac{13}{8}\right) = \frac{\frac{91 + 32}{8}}{\frac{65 - 24}{8}} = \frac{\frac{123}{8}}{\frac{41}{8}} = \frac{123}{41} = 3 \] Step 4: Conclusion.
Thus, \( (g \circ f)(3) = 3 \), and the correct answer is \( \boxed{3} \).