f(x) = 4x + 3 and g(x) = 2 - 3x find (g.f)(3)

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 \(g(x)\).

Learn how to find (g ∘ f)(3) using function composition with detailed step-by-step solutions. This problem involves evaluating functions f(x) = 4x + 3 and g(x) = 2 - 3x at x = 3. Suitable for Grades 10-12.

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