If `f(x)=4x-3` and `g(x)=x-4`, which of the following has a value of `-11` ?

Options

A. `f(g(2))`
B. `g(f(2))`
C. `g(f(3))`
D. `f(g(3))`

Explanation

To solve the problem, we need to evaluate the functions \( f(x) = 4x - 3 \) and \( g(x) = x - 4 \) for different options to find which one equals \(-11\).

### Step-by-step Solution:

1. **Evaluate Option A: \( f(g(2)) \)**
- First, calculate \( g(2) \):
\[ g(2) = 2 - 4 = -2 \]
- Now, substitute \( g(2) \) into \( f \):
\[ f(g(2)) = f(-2) = 4(-2) - 3 = -8 - 3 = -11 \]
- Thus, \( f(g(2)) = -11 \).

2. **Evaluate Option B: \( g(f(2)) \)**
- First, calculate \( f(2) \):
\[ f(2) = 4(2) - 3 = 8 - 3 = 5 \]
- Now, substitute \( f(2) \) into \( g \):
\[ g(f(2)) = g(5) = 5 - 4 = 1 \]
- Thus, \( g(f(2)) = 1 \).

3. **Evaluate Option C: \( g(f(3)) \)**
- First, calculate \( f(3) \):
\[ f(3) = 4(3) - 3 = 12 - 3 = 9 \]
- Now, substitute \( f(3) \) into \( g \):
\[ g(f(3)) = g(9) = 9 - 4 = 5 \]
- Thus, \( g(f(3)) = 5 \).

4. **Evaluate Option D: \( f(g(3)) \)**
- First, calculate \( g(3) \):
\[ g(3) = 3 - 4 = -1 \]
- Now, substitute \( g(3) \) into \( f \):
\[ f(g(3)) = f(-1) = 4(-1) - 3 = -4 - 3 = -7 \]
- Thus, \( f(g(3)) = -7 \).

### Summary of Results:
- Option A: \( f(g(2)) = -11 \)
- Option B: \( g(f(2)) = 1 \)
- Option C: \( g(f(3)) = 5 \)
- Option D: \( f(g(3)) = -7 \)

The only option that has a value of \(-11\) is **Option A**: \( f(g(2)) \).