To find the inverse of the function \( f(x) = -4x + 12 \), we need to follow a series of steps. Let's go through them one by one.
1. Step 1: Start by expressing \( f(x) \) as \( y \):
[tex]\[
y = -4x + 12
\][/tex]
2. Step 2: Swap \( x \) and \( y \):
[tex]\[
x = -4y + 12
\][/tex]
3. Step 3: Solve for \( y \) in terms of \( x \):
- First, isolate the term involving \( y \) by subtracting 12 from both sides:
[tex]\[
x - 12 = -4y
\][/tex]
- Next, divide both sides by \(-4\):
[tex]\[
y = \frac{x - 12}{-4}
\][/tex]
- Simplify the fraction:
[tex]\[
y = -\frac{x}{4} + \frac{12}{4}
\][/tex]
- Further simplification gives:
[tex]\[
y = -\frac{x}{4} + 3
\][/tex]
4. Step 4: Replace \( y \) with \( f^{-1}(x) \):
[tex]\[
f^{-1}(x) = -\frac{x}{4} + 3
\][/tex]
So, the inverse function \( f^{-1}(x) \) is:
[tex]\[
f^{-1}(x) = -\frac{x}{4} + 3
\][/tex]
To fit it into the format:
[tex]\[
f^{-1}(x) = -\frac{x}{[4]} + [3]
\][/tex]
The denominator is 4, and the constant is 3. Therefore, the values to fill in are:
[tex]\[
4 \text{ and } 3
\][/tex]
Hence, the answer is:
[tex]\[
f^{-1}(x) = -\frac{x}{4} + 3
\][/tex]
