To find $(g \circ f)(3)$, we need to evaluate the composition of the functions $g(x)$ and $f(x)$ at $x = 3$. This composition is defined as:

$(g \circ f)(x) = g(f(x))$

Given Functions

• $f(x) = 4x + 3$
• $g(x) = 2 - 3x$

Step-by-Step Solution

1. Calculate $f(3)$:

   Substitute $x = 3$ into $f(x)$:

   $f(3) = 4(3) + 3$ $f(3) = 12 + 3$ $f(3) = 15$

2. Calculate $g(f(3)) = g(15)$:

   Now substitute $f(3) = 15$ into $g(x)$:

   $g(15) = 2 - 3(15)$ $g(15) = 2 - 45$ $g(15) = -43$

Conclusion

$(g \circ f)(3) = -43$

Would you like more details or have any questions about this solution?

Related Questions:

1. What is $(f \circ g)(3)$ given the functions above?
2. How do you find $(g \circ g)(x)$ for any $x$?
3. What is the slope and y-intercept of the functions $f(x)$ and $g(x)$?
4. How can you find the inverse of $g(x) = 2 - 3x$?
5. How would you solve $(g \circ f)(x) = 0$?

Tip:

To compute $(g \circ f)(x)$, always evaluate $f(x)$ first, then substitute its value into $g(x)$.