Find `k` if `fog=gof` where `f(x)=3x-1`, `g(x)=4x+k`.

Answer

To find the value of \( k \) such that \( f(g(x)) = g(f(x)) \), where \( f(x) = 3x - 1 \) and \( g(x) = 4x + k \), we will follow these steps:

### Step 1: Compute \( f(g(x)) \)

First, we need to substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(4x + k) = 3(4x + k) - 1 \]
Now, simplify this expression:
\[ = 12x + 3k - 1 \]

### Step 2: Compute \( g(f(x)) \)

Next, we substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(3x - 1) = 4(3x - 1) + k \]
Now, simplify this expression:
\[ = 12x - 4 + k \]

### Step 3: Set the two expressions equal to each other

Since we have \( f(g(x)) = g(f(x)) \), we can set the two expressions from Steps 1 and 2 equal:
\[ 12x + 3k - 1 = 12x - 4 + k \]

### Step 4: Eliminate \( 12x \) from both sides

Subtract \( 12x \) from both sides:
\[ 3k - 1 = -4 + k \]

### Step 5: Solve for \( k \)

Now, rearranging the equation to isolate \( k \):
\[ 3k - k = -4 + 1 \]
\[ 2k = -3 \]
\[ k = -\frac{3}{2} \]

### Final Answer

Thus, the value of \( k \) is:
\[ \boxed{-\frac{3}{2}} \]

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